Algebraic number field

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vector-toc-level-1"> <a class="vector-toc-link" href="#Algebraicity,_and_ring_of_integers"> <div class="vector-toc-text"> 3 Algebraicity, and ring of integers </div> </a> <button aria-controls="toc-Algebraicity,_and_ring_of_integers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Algebraicity, and ring of integers subsection </button> <ul id="toc-Algebraicity,_and_ring_of_integers-sublist" class="vector-toc-list"> <li id="toc-Unique_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unique_factorization"> <div class="vector-toc-text"> 3.1 Unique factorization </div> </a> <ul id="toc-Unique_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Analytic_objects:_ζ-functions,_L-functions,_and_class_number_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Analytic_objects:_ζ-functions,_L-functions,_and_class_number_formula"> <div class="vector-toc-text"> 3.2 Analytic objects: ζ-functions, L-functions, and class number formula </div> </a> <ul id="toc-Analytic_objects:_ζ-functions,_L-functions,_and_class_number_formula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bases_for_number_fields" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bases_for_number_fields"> <div class="vector-toc-text"> 4 Bases for number fields </div> </a> <button aria-controls="toc-Bases_for_number_fields-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Bases for number fields subsection </button> <ul id="toc-Bases_for_number_fields-sublist" class="vector-toc-list"> <li id="toc-Integral_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_basis"> <div class="vector-toc-text"> 4.1 Integral basis </div> </a> <ul id="toc-Integral_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_basis"> <div class="vector-toc-text"> 4.2 Power basis </div> </a> <ul id="toc-Power_basis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Regular_representation,_trace_and_discriminant" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Regular_representation,_trace_and_discriminant"> <div class="vector-toc-text"> 5 Regular representation, trace and discriminant </div> </a> <button aria-controls="toc-Regular_representation,_trace_and_discriminant-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Regular representation, trace and discriminant subsection </button> <ul id="toc-Regular_representation,_trace_and_discriminant-sublist" class="vector-toc-list"> <li id="toc-Example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example"> <div class="vector-toc-text"> 5.1 Example </div> </a> <ul id="toc-Example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 5.2 Properties </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_with_integral_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Example_with_integral_basis"> <div class="vector-toc-text"> 5.3 Example with integral basis </div> </a> <ul id="toc-Example_with_integral_basis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Places" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Places"> <div class="vector-toc-text"> 6 Places </div> </a> <button aria-controls="toc-Places-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Places subsection </button> <ul id="toc-Places-sublist" class="vector-toc-list"> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> 6.1 Examples </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Archimedean_places" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Archimedean_places"> <div class="vector-toc-text"> 6.2 Archimedean places </div> </a> <ul id="toc-Archimedean_places-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-Archimedean_or_ultrametric_places" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-Archimedean_or_ultrametric_places"> <div class="vector-toc-text"> 6.3 Non-Archimedean or ultrametric places </div> </a> <ul id="toc-Non-Archimedean_or_ultrametric_places-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Prime_ideals_in_OK" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Prime_ideals_in_OK"> <div class="vector-toc-text"> 6.4 Prime ideals in OK </div> </a> <ul id="toc-Prime_ideals_in_OK-sublist" class="vector-toc-list"> <li id="toc-Lying_over_theorem_and_places" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Lying_over_theorem_and_places"> <div class="vector-toc-text"> 6.4.1 Lying over theorem and places </div> </a> <ul id="toc-Lying_over_theorem_and_places-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Ramification" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ramification"> <div class="vector-toc-text"> 7 Ramification </div> </a> <button aria-controls="toc-Ramification-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Ramification subsection </button> <ul id="toc-Ramification-sublist" class="vector-toc-list"> <li id="toc-An_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#An_example"> <div class="vector-toc-text"> 7.1 An example </div> </a> <ul id="toc-An_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dedekind_discriminant_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dedekind_discriminant_theorem"> <div class="vector-toc-text"> 7.2 Dedekind discriminant theorem </div> </a> <ul id="toc-Dedekind_discriminant_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Galois_groups_and_Galois_cohomology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Galois_groups_and_Galois_cohomology"> <div class="vector-toc-text"> 8 Galois groups and Galois cohomology </div> </a> <ul id="toc-Galois_groups_and_Galois_cohomology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local-global_principle" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Local-global_principle"> <div class="vector-toc-text"> 9 Local-global principle </div> </a> <button aria-controls="toc-Local-global_principle-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Local-global principle subsection </button> <ul id="toc-Local-global_principle-sublist" class="vector-toc-list"> <li id="toc-Local_and_global_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Local_and_global_fields"> <div class="vector-toc-text"> 9.1 Local and global fields </div> </a> <ul id="toc-Local_and_global_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hasse_principle" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hasse_principle"> <div class="vector-toc-text"> 9.2 Hasse principle </div> </a> <ul id="toc-Hasse_principle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adeles_and_ideles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adeles_and_ideles"> <div class="vector-toc-text"> 9.3 Adeles and ideles </div> </a> <ul id="toc-Adeles_and_ideles-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 10 See also </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle See also subsection </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 10.1 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_number_theory"> <div class="vector-toc-text"> 10.2 Algebraic number theory </div> </a> <ul id="toc-Algebraic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Class_field_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Class_field_theory"> <div class="vector-toc-text"> 10.3 Class field theory </div> </a> <ul id="toc-Class_field_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 11 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Algebraic number field</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 36 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-36" aria-hidden="true" > 36 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%82%D9%84_%D8%A7%D9%84%D8%A3%D8%B9%D8%AF%D8%A7%D8%AF_%D8%A7%D9%84%D8%AC%D8%A8%D8%B1%D9%8A%D8%A9" title="حقل الأعداد الجبرية – Arabic" lang="ar" hreflang="ar" data-title="حقل الأعداد الجبرية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Cuerpu_de_n%C3%BAmberos_alxebraicos" title="Cuerpu de númberos alxebraicos – Asturian" lang="ast" hreflang="ast" data-title="Cuerpu de númberos alxebraicos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D2%BB%D0%B0%D0%BD%D0%B4%D0%B0%D1%80_%D1%8F%D0%BB%D0%B0%D0%BD%D1%8B" title="Алгебраик һандар яланы – Bashkir" lang="ba" hreflang="ba" data-title="Алгебраик һандар яланы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D0%B0%D0%B5_%D0%BB%D1%96%D0%BA%D0%B0%D0%B2%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Алгебраічнае лікавае поле – Belarusian" lang="be" hreflang="be" data-title="Алгебраічнае лікавае поле" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Cos_dels_nombres_algebraics" title="Cos dels nombres algebraics – Catalan" lang="ca" hreflang="ca" data-title="Cos dels nombres algebraics" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C4%8C%C3%ADseln%C3%A9_t%C4%9Bleso" title="Číselné těleso – Czech" lang="cs" hreflang="cs" data-title="Číselné těleso" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebraischer_Zahlk%C3%B6rper" title="Algebraischer Zahlkörper – German" lang="de" hreflang="de" data-title="Algebraischer Zahlkörper" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Algebraline_arvukorpus" title="Algebraline arvukorpus – Estonian" lang="et" hreflang="et" data-title="Algebraline arvukorpus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%8E%CE%BC%CE%B1_%CE%91%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD" title="Σώμα Αριθμών – Greek" lang="el" hreflang="el" data-title="Σώμα Αριθμών" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuerpo_de_n%C3%BAmeros_algebraicos" title="Cuerpo de números algebraicos – Spanish" lang="es" hreflang="es" data-title="Cuerpo de números algebraicos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbakien_eremu_aljebraikoa" title="Zenbakien eremu aljebraikoa – Basque" lang="eu" hreflang="eu" data-title="Zenbakien eremu aljebraikoa" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86_%D8%AC%D8%A8%D8%B1%DB%8C_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF" title="میدان جبری اعداد – Persian" lang="fa" hreflang="fa" data-title="میدان جبری اعداد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Corps_de_nombres" title="Corps de nombres – French" lang="fr" hreflang="fr" data-title="Corps de nombres" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Corpo_de_n%C3%BAmeros_alx%C3%A9bricos" title="Corpo de números alxébricos – Galician" lang="gl" hreflang="gl" data-title="Corpo de números alxébricos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EC%88%98%EC%A0%81_%EC%88%98%EC%B2%B4" title="대수적 수체 – Korean" lang="ko" hreflang="ko" data-title="대수적 수체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="बीजगणितीय संख्या सिद्धान्त – Hindi" lang="hi" hreflang="hi" data-title="बीजगणितीय संख्या सिद्धान्त" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Medan_bilangan_aljabar" title="Medan bilangan aljabar – Indonesian" lang="id" hreflang="id" data-title="Medan bilangan aljabar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Campo_di_numeri" title="Campo di numeri – Italian" lang="it" hreflang="it" data-title="Campo di numeri" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%93%D7%94_%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D" title="שדה מספרים – Hebrew" lang="he" hreflang="he" data-title="שדה מספרים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra%C3%AFsch_getallenlichaam" title="Algebraïsch getallenlichaam – Dutch" lang="nl" hreflang="nl" data-title="Algebraïsch getallenlichaam" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E4%BD%93" title="代数体 – Japanese" lang="ja" hreflang="ja" data-title="代数体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Talkropp" title="Talkropp – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Talkropp" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cia%C5%82o_liczbowe" title="Ciało liczbowe – Polish" lang="pl" hreflang="pl" data-title="Ciało liczbowe" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Corpo_de_n%C3%BAmeros_alg%C3%A9bricos" title="Corpo de números algébricos – Portuguese" lang="pt" hreflang="pt" data-title="Corpo de números algébricos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D0%BE%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Алгебраическое числовое поле – Russian" lang="ru" hreflang="ru" data-title="Алгебраическое числовое поле" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Number_field" title="Number field – Simple English" lang="en-simple" hreflang="en-simple" data-title="Number field" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Obseg_algebrskih_%C5%A1tevil" title="Obseg algebrskih števil – Slovenian" lang="sl" hreflang="sl" data-title="Obseg algebrskih števil" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D1%99%D0%B5_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D0%B0%D1%80%D1%81%D0%BA%D0%B8%D1%85_%D0%B1%D1%80%D0%BE%D1%98%D0%B5%D0%B2%D0%B0" title="Поље алгебарских бројева – Serbian" lang="sr" hreflang="sr" data-title="Поље алгебарских бројева" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a> <dl><dd>• <a href="/wiki/Subring" title="Subring">Subrings</a></dd> <dd>• <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd> <dd>• <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a> <dl><dd>• <a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd> <dd>• <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd> <dd>• <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></dd> <dd>• <a href="/wiki/Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd> <dd>• <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl> <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a> <dl><dd>• <a href="/wiki/Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd> <dd>• <a href="/wiki/Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd> <dd>• <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl> <a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> <dl><dd>• <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></dd> <dd>• <a href="/wiki/Associative_algebra" title="Associative algebra">Associative algebra</a></dd> <dd>• <a href="/wiki/Graded_ring" title="Graded ring">Graded ring</a></dd> <dd>• <a href="/wiki/Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd> <dd>• <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a> <dl><dd>• <a href="/wiki/Integer" title="Integer">Initial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ab495cb8cfbac68a9322af662c3d6c7dbe494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"></dd></dl></dd></dl> Related structures <dl><dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a> <dl><dd>• <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></dd> <dd>• <a href="/wiki/Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd> <dd>• <a href="/wiki/Semiring" title="Semiring">Semiring</a> <dl><dd>• <a href="/wiki/Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a> <dl><dd>• <a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a> <dl><dd>• <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd> <dd>• <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a></dd> <dd>• <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd> <dd>• <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd> <dd>• <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd> <dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd> <dd>• <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl> <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> <dl><dd>• <a class="mw-selflink selflink">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo n</a></dd> <dd>• <a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></dd> <dd>• <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer p-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14af623e08c241266c125ad927dd35086ec8ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}"></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a> <dl><dd>• <a href="/wiki/Division_ring" title="Division ring">Division ring</a></dd> <dd>• <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd> <dd>• <a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a></dd> <dd>• <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl> <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a> <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a> <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Ring_theory_sidebar" title="Template:Ring theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ring_theory_sidebar" title="Template talk:Ring theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ring_theory_sidebar" title="Special:EditPage/Template:Ring theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an algebraic number field (or simply number field) is an <a href="/wiki/Extension_field" class="mw-redirect" title="Extension field">extension field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> of the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> such that the <a href="/wiki/Field_extension" title="Field extension">field extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6782a2935122dafec2947d1d7ba168be4399ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }"> has <a href="/wiki/Degree_of_a_field_extension" title="Degree of a field extension">finite degree</a> (and hence is an <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic</a> field extension). Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is a field that contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> and has finite <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">dimension</a> when considered as a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">. The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. This study reveals hidden structures behind the rational numbers, by using algebraic methods. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Prerequisites">Prerequisites</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=2" title="Edit section: Prerequisites">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> and <a href="/wiki/Vector_space" title="Vector space">Vector space</a></div> The notion of algebraic number field relies on the concept of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. A field consists of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of elements together with two operations, namely <a href="/wiki/Addition" title="Addition">addition</a>, and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, and some <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a> assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication. A prominent example of a field is the field of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, commonly denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">, together with its usual operations of addition and multiplication. Another notion needed to define algebraic number fields is <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>. To the extent needed here, vector spaces can be thought of as consisting of sequences (or <a href="/wiki/Tuple" title="Tuple">tuples</a>) <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},\dots )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1fabaef4eaa2225bb42a4573912d997ef34c4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},\dots )}"></dd></dl> whose entries are elements of a fixed field, such as the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">. Any two such sequences can be added by adding the corresponding entries. Furthermore, all members of any sequence can be multiplied by a single element c of the fixed field. These two operations known as <a href="/wiki/Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a> and <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "<a href="/wiki/Infinite-dimensional" class="mw-redirect" title="Infinite-dimensional">infinite-dimensional</a>", that is to say that the sequences constituting the vector spaces may be of infinite length. If, however, the vector space consists of finite sequences <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56204ada1ecf9dd5c6beddfdfd0f341cd69ff632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}">,</dd></dl> the vector space is said to be of finite <a href="/wiki/Hamel_dimension" class="mw-redirect" title="Hamel dimension">dimension</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. <div class="mw-heading mw-heading3"><h3 id="Definition_2"> Definition</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=3" title="Edit section: Definition">edit</a>]</div> An algebraic number field (or simply number field) is a finite-<a href="/wiki/Degree_of_a_field_extension" title="Degree of a field extension">degree</a> <a href="/wiki/Field_extension" title="Field extension">field extension</a> of the field of rational numbers. Here degree means the dimension of the field as a vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">. <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=4" title="Edit section: Examples">edit</a>]</div> <ul><li>The smallest and most basic number field is the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> of rational numbers. Many properties of general number fields are modeled after the properties of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">. At the same time, many other properties of algebraic number fields are substantially different from the properties of rational numbers—one notable example is that the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a> of a number field is not a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>, in general.</li> <li>The <a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (i)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aace8b46c21e37def400ec270519b9133b498b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.42ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (i)}"> (read as "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> <a href="/wiki/Adjunction_(field_theory)" class="mw-redirect" title="Adjunction (field theory)">adjoined</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">"), form the first (historically) non-trivial example of a number field. Its elements are elements of the form <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+bi}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+bi}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92f853c2c9235c06be640b91b7c75e2a907cbda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.87ex; height:2.343ex;" alt="{\displaystyle a+bi}"> where both a and b are rational numbers and i is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ca91534578075175b2088069fb6a9a414eaa7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.573ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1.}"> Explicitly, for real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c,d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c,d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd0b3d3b09ae6ae430f09c4b317742c56e8acace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.552ex; height:2.509ex;" alt="{\displaystyle a,b,c,d}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}(a+bi)+(c+di)&=&(a+c)+(b+d)i\\(a+bi)\cdot (c+di)&=&(ac-bd)+(ad+bc)i\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mi>i</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−</mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>d</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mi>i</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}(a+bi)+(c+di)&=&(a+c)+(b+d)i\\(a+bi)\cdot (c+di)&=&(ac-bd)+(ad+bc)i\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14b638c9175554f2733c57870d9fc3265436cf37" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.242ex; height:6.176ex;" alt="{\displaystyle {\begin{matrix}(a+bi)+(c+di)&=&(a+c)+(b+d)i\\(a+bi)\cdot (c+di)&=&(ac-bd)+(ad+bc)i\end{matrix}}}"> Non-zero Gaussian rational numbers are <a href="/wiki/Invertible" class="mw-redirect" title="Invertible">invertible</a>, which can be seen from the identity <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+bi)\left({\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i\right)={\frac {(a+bi)(a-bi)}{a^{2}+b^{2}}}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+bi)\left({\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i\right)={\frac {(a+bi)(a-bi)}{a^{2}+b^{2}}}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4958691dec3a538222319c3768dd68ce73abd540" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:55.357ex; height:6.343ex;" alt="{\displaystyle (a+bi)\left({\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i\right)={\frac {(a+bi)(a-bi)}{a^{2}+b^{2}}}=1.}"> It follows that the Gaussian rationals form a number field that is two-dimensional as a vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">.</li> <li>More generally, for any <a href="/wiki/Square-free" class="mw-redirect" title="Square-free">square-free</a> integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}">, the <a href="/wiki/Quadratic_field" title="Quadratic field">quadratic field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ({\sqrt {d}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ({\sqrt {d}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9757d116555dd605228d352039f5f491d2967c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.769ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} ({\sqrt {d}})}"> is a number field obtained by adjoining the square root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"> to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d51c156019d41a34e4ab088ef479980c6c6e5614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.285ex; height:2.343ex;" alt="{\displaystyle d=-1}">.</li> <li><a href="/wiki/Cyclotomic_field" title="Cyclotomic field">The cyclotomic field</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (\zeta _{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\zeta _{n}),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9580400ada75abfe19b9e9af59321521a010899" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\zeta _{n}),}">where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta _{n}=\exp {(2\pi i/n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{n}=\exp {(2\pi i/n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb602eb1263be3fcb5fb7606addbe771e963b84d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.938ex; height:2.843ex;" alt="{\displaystyle \zeta _{n}=\exp {(2\pi i/n)}}">, is a number field obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> by adjoining a primitive <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">th <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e41a78e5c2df8ff2fab3863a0f10fa7539b973c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.237ex; height:2.509ex;" alt="{\displaystyle \zeta _{n}}">. This field contains all complex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">th roots of unity and its dimension over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f067864064667dd5f8b2508b9cbf983d89788629" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"> is the <a href="/wiki/Euler_totient_function" class="mw-redirect" title="Euler totient function">Euler totient function</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Non-examples">Non-examples</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=5" title="Edit section: Non-examples">edit</a>]</div> <ul><li>The <a href="/wiki/Real_number" title="Real number">real numbers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, and the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">, are fields that have infinite dimension as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">-vector spaces; hence, they are not number fields. This follows from the <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountability</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"> as sets, whereas every number field is necessarily <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a>.</li> <li>The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34ada693d0caa6a18f306839c57b2d78569312f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.862ex; height:3.009ex;" alt="{\displaystyle \mathbb {Q} ^{2}}"> of <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pairs</a> of rational numbers, with the entry-wise addition and multiplication is a two-dimensional <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">. However, it is not a field, since it has <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisors</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0)\cdot (0,1)=(1\cdot 0,0\cdot 1)=(0,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>⋅</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,0)\cdot (0,1)=(1\cdot 0,0\cdot 1)=(0,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daf909017fa6b53abbadf42dbc1b4b6074804b18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.232ex; height:2.843ex;" alt="{\displaystyle (1,0)\cdot (0,1)=(1\cdot 0,0\cdot 1)=(0,0)}"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Algebraicity,_and_ring_of_integers">Algebraicity, and ring of integers</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=6" title="Edit section: Algebraicity, and ring of integers">edit</a>]</div> Generally, in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, a field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb79e236fa3e355b2c2e37b37ccf84ad3f9dabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle K/L}"> is <a href="/wiki/Algebraic_field_extension" class="mw-redirect" title="Algebraic field extension">algebraic</a> if every element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> of the bigger field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is the zero of a (nonzero) <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> with coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{0},\ldots ,e_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{0},\ldots ,e_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fb082ab1407746e9b8d06b23ce9f10d30dbe57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.074ex; height:2.009ex;" alt="{\displaystyle e_{0},\ldots ,e_{m}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(f)=e_{m}f^{m}+e_{m-1}f^{m-1}+\cdots +e_{1}f+e_{0}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(f)=e_{m}f^{m}+e_{m-1}f^{m-1}+\cdots +e_{1}f+e_{0}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70595ddb355ebb010e2d601081eb259c45d4e96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:47.054ex; height:3.176ex;" alt="{\displaystyle p(f)=e_{m}f^{m}+e_{m-1}f^{m-1}+\cdots +e_{1}f+e_{0}=0}"></dd></dl> Every field extension of finite degree is algebraic. (Proof: for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, simply consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,x,x^{2},x^{3},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,x,x^{2},x^{3},\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db592780a2a4ae86f4f12e7894d4e75381323bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.119ex; height:3.009ex;" alt="{\displaystyle 1,x,x^{2},x^{3},\ldots }"> – we get a linear dependence, i.e. a polynomial that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is a root of.) In particular this applies to algebraic number fields, so any element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> of an algebraic number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> can be written as a zero of a polynomial with rational coefficients. Therefore, elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> are also referred to as <a href="/wiki/Algebraic_numbers" class="mw-redirect" title="Algebraic numbers">algebraic numbers</a>. Given a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(f)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(f)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad359a461e0e36aa53ce18fecad90f01d0577843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:8.608ex; height:2.843ex;" alt="{\displaystyle p(f)=0}">, it can be arranged such that the leading coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d550719ca5dca131b4b82d021c218db9ea262ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.759ex; height:2.009ex;" alt="{\displaystyle e_{m}}"> is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a>. In general it will have rational coefficients. If, however, the monic polynomial's coefficients are actually all integers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is called an <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integer</a>. Any (usual) integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89579fe9cb330174ba72262b85764c9bcd827b57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.479ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {Z} }"> is an algebraic integer, as it is the zero of the linear monic polynomial: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)=t-z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)=t-z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba94c6e8dfcfdad4c86fac0555f0d281ccf291db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:11.774ex; height:2.843ex;" alt="{\displaystyle p(t)=t-z}">.</dd></dl> It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated module</a>, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> form a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> called the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. It is a <a href="/wiki/Subring" title="Subring">subring</a> of (that is, a ring contained in) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. A field contains no <a href="/wiki/Zero_divisors" class="mw-redirect" title="Zero divisors">zero divisors</a> and this property is inherited by any subring, so the ring of integers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>. The field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of the integral domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. This way one can get back and forth between the algebraic number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> and its ring of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. Rings of algebraic integers have three distinctive properties: firstly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> is an integral domain that is <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">integrally closed</a> in its field of fractions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. Secondly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> is a <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian ring</a>. Finally, every nonzero <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> is <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal</a> or, equivalently, the <a href="/wiki/Krull_dimension" title="Krull dimension">Krull dimension</a> of this ring is one. An abstract commutative ring with these three properties is called a <a href="/wiki/Dedekind_ring" class="mw-redirect" title="Dedekind ring">Dedekind ring</a> (or Dedekind domain), in honor of <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>, who undertook a deep study of rings of algebraic integers. <div class="mw-heading mw-heading3"><h3 id="Unique_factorization">Unique factorization</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=7" title="Edit section: Unique factorization">edit</a>]</div> For general <a href="/wiki/Dedekind_ring" class="mw-redirect" title="Dedekind ring">Dedekind rings</a>, in particular rings of integers, there is a unique factorization of <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> into a product of <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a>. For example, the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (6)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2cb61a8d9601228914dbfe57bbb933741acb5a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (6)}"> in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} [{\sqrt {-5}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} [{\sqrt {-5}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff18a58b48d228e03878fada8601ee4cc78fa617" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:3.009ex;" alt="{\displaystyle \mathbf {Z} [{\sqrt {-5}}]}"> of <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a> factors into prime ideals as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (6)=(2,1+{\sqrt {-5}})(2,1-{\sqrt {-5}})(3,1+{\sqrt {-5}})(3,1-{\sqrt {-5}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (6)=(2,1+{\sqrt {-5}})(2,1-{\sqrt {-5}})(3,1+{\sqrt {-5}})(3,1-{\sqrt {-5}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6fb0abc517252b1155f5a7261f39636a7fc8781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.73ex; height:3.009ex;" alt="{\displaystyle (6)=(2,1+{\sqrt {-5}})(2,1-{\sqrt {-5}})(3,1+{\sqrt {-5}})(3,1-{\sqrt {-5}})}"></dd></dl> However, unlike <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776aaf12c2da4b78ca777cb8295c2000bfd51f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.634ex; height:2.176ex;" alt="{\displaystyle \mathbf {Z} }"> as the ring of integers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }">, the ring of integers of a proper extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132d0144479d6f47c30ad82a65d458966ccbe928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.008ex; height:2.509ex;" alt="{\displaystyle \mathbf {Q} }"> need not admit <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization</a> of numbers into a product of prime numbers or, more precisely, <a href="/wiki/Prime_element" title="Prime element">prime elements</a>. This happens already for <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a>, for example in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}=\mathbf {Z} [{\sqrt {-5}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}=\mathbf {Z} [{\sqrt {-5}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c7679d6634d4f3ba8397144e5540e806e8160ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.183ex; height:3.343ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}=\mathbf {Z} [{\sqrt {-5}}]}">, the uniqueness of the factorization fails: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})\cdot (1-{\sqrt {-5}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})\cdot (1-{\sqrt {-5}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/716b97efb68ed14703c7d72c40511e4508f5ef1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.48ex; height:3.009ex;" alt="{\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})\cdot (1-{\sqrt {-5}})}"></dd></dl> Using the <a href="/wiki/Field_norm" title="Field norm">norm</a> it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f29da7f6346d44ab9c0af41d278f5cd567a724c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.251ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}}">. <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domains</a> are unique factorization domains; for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} [i]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} [i]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a617cf5867f951fefb72f3ab7278e0f6f1eedd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.73ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} [i]}">, the ring of <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} [\omega ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>ω</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} [\omega ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0932b22f32aa89b10ed7ae60a65c5c3250ed01c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.373ex; height:2.843ex;" alt="{\displaystyle \mathbf {Z} [\omega ]}">, the ring of <a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> is a cube root of unity (unequal to 1), have this property.<a href="#cite_note-1">[1]</a> <div class="mw-heading mw-heading3"><h3 id="Analytic_objects:_ζ-functions,_L-functions,_and_class_number_formula">Analytic objects: ζ-functions, L-functions, and class number formula</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=8" title="Edit section: Analytic objects: ζ-functions, L-functions, and class number formula">edit</a>]</div> The failure of unique factorization is measured by the <a href="/wiki/Class_number_(number_theory)" class="mw-redirect" title="Class number (number theory)">class number</a>, commonly denoted h, the cardinality of the so-called <a href="/wiki/Ideal_class_group" title="Ideal class group">ideal class group</a>. This group is always finite. The ring of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> possesses unique factorization if and only if it is a principal ring or, equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> has <a href="/wiki/List_of_number_fields_with_class_number_one" title="List of number fields with class number one">class number 1</a>. Given a number field, the class number is often difficult to compute. The <a href="/wiki/Class_number_problem" title="Class number problem">class number problem</a>, going back to <a href="/wiki/Gauss" class="mw-redirect" title="Gauss">Gauss</a>, is concerned with the existence of imaginary quadratic number fields (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ({\sqrt {-d}}),d\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mi>d</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mi>d</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ({\sqrt {-d}}),d\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92c399b02d24a89ace87df32e9ffb53e3ae7dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.288ex; height:3.343ex;" alt="{\displaystyle \mathbf {Q} ({\sqrt {-d}}),d\geq 1}">) with prescribed class number. The <a href="/wiki/Class_number_formula" title="Class number formula">class number formula</a> relates h to other fundamental invariants of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. It involves the <a href="/wiki/Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta _{K}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{K}(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3551683bc349c27410ef87c8443a695b62e5996c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.611ex; height:2.843ex;" alt="{\displaystyle \zeta _{K}(s)}">, a function in a complex variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}">, defined by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta _{K}(s):=\prod _{\mathfrak {p}}{\frac {1}{1-N({\mathfrak {p}})^{-s}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{K}(s):=\prod _{\mathfrak {p}}{\frac {1}{1-N({\mathfrak {p}})^{-s}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20a1e5b8f3ea554385820d368e343042d8590c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.516ex; height:6.676ex;" alt="{\displaystyle \zeta _{K}(s):=\prod _{\mathfrak {p}}{\frac {1}{1-N({\mathfrak {p}})^{-s}}}.}"></dd></dl> (The product is over all prime ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N({\mathfrak {p}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N({\mathfrak {p}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce70de1939a3070ef6905f543b1bf34aa843df08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.035ex; height:2.843ex;" alt="{\displaystyle N({\mathfrak {p}})}"> denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the <a href="/wiki/Residue_field" title="Residue field">residue field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}/{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}/{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c556eca07640ac705e342406fc3478aef93e4cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.868ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{K}/{\mathfrak {p}}}">. The infinite product converges only for <a href="/wiki/Real_part" class="mw-redirect" title="Real part">Re</a>(s) > 1; in general <a href="/wiki/Analytic_continuation" title="Analytic continuation">analytic continuation</a> and the <a href="/wiki/Functional_equation" title="Functional equation">functional equation</a> for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the <a href="/wiki/Riemann_zeta-function" class="mw-redirect" title="Riemann zeta-function">Riemann zeta-function</a> in that ζ<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">(s) = ζ(s). The class number formula states that ζ<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">(s) has a <a href="/wiki/Simple_pole" class="mw-redirect" title="Simple pole">simple pole</a> at s = 1 and at this point the <a href="/wiki/Residue_(complex_analysis)" title="Residue (complex analysis)">residue</a> is given by <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot h\cdot \operatorname {Reg} }{w\cdot {\sqrt {|D|}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>⋅</mo> <mi>h</mi> <mo>⋅</mo> <mi>Reg</mi> </mrow> <mrow> <mi>w</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot h\cdot \operatorname {Reg} }{w\cdot {\sqrt {|D|}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15be5a7877e33b04317e52fadbf6326684d82ab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.842ex; height:7.009ex;" alt="{\displaystyle {\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot h\cdot \operatorname {Reg} }{w\cdot {\sqrt {|D|}}}}.}"></dd></dl> Here r1 and r2 classically denote the number of <a href="/wiki/Real_and_complex_embeddings" class="mw-redirect" title="Real and complex embeddings">real embeddings</a> and pairs of <a href="/wiki/Real_and_complex_embeddings" class="mw-redirect" title="Real and complex embeddings">complex embeddings</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, respectively. Moreover, Reg is the <a href="/wiki/Regulator_(mathematics)" class="mw-redirect" title="Regulator (mathematics)">regulator</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, w the number of <a href="/wiki/Root_of_unity" title="Root of unity">roots of unity</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> and D is the discriminant of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\chi ,s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>χ</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\chi ,s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2ede56cb317a2561bcfa48a1ea433b274781bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.972ex; height:2.843ex;" alt="{\displaystyle L(\chi ,s)}"> are a more refined variant of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta (s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd45922057e4d7a5718ce5ed703ab493c63897a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.995ex; height:2.843ex;" alt="{\displaystyle \zeta (s)}">. Both types of functions encode the arithmetic behavior of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, respectively. For example, <a href="/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem</a> asserts that in any <a href="/wiki/Arithmetic_progression" title="Arithmetic progression">arithmetic progression</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,a+m,a+2m,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>m</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>m</mi> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,a+m,a+2m,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce8b1bf86d72b63c7b26631f29a88e0471c147aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.439ex; height:2.509ex;" alt="{\displaystyle a,a+m,a+2m,\ldots }"></dd></dl> with <a href="/wiki/Coprime" class="mw-redirect" title="Coprime">coprime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">, there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">-function is nonzero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac386d8f227fb823cede9b3e33d706cad3ed306" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}">. Using much more advanced techniques including <a href="/wiki/Algebraic_K-theory" title="Algebraic K-theory">algebraic K-theory</a> and <a href="/wiki/Tamagawa_measure" class="mw-redirect" title="Tamagawa measure">Tamagawa measures</a>, modern number theory deals with a description, if largely conjectural (see <a href="/wiki/Tamagawa_number_conjecture" class="mw-redirect" title="Tamagawa number conjecture">Tamagawa number conjecture</a>), of values of more general <a href="/wiki/L-function" title="L-function">L-functions</a>.<a href="#cite_note-2">[2]</a> <div class="mw-heading mw-heading2"><h2 id="Bases_for_number_fields">Bases for number fields</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=9" title="Edit section: Bases for number fields">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Integral_basis">Integral basis</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=10" title="Edit section: Integral basis">edit</a>]</div> An <a href="/wiki/Integral_basis" class="mw-redirect" title="Integral basis">integral basis</a> for a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is a set <dl><dd>B = {b1, …, bn}</dd></dl> of n algebraic integers in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> such that every element of the ring of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> can be written uniquely as a Z-linear combination of elements of B; that is, for any x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> we have <dl><dd>x = m1b1 + ⋯ + mnbn,</dd></dl> where the mi are (ordinary) integers. It is then also the case that any element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> can be written uniquely as <dl><dd>m1b1 + ⋯ + mnbn,</dd></dl> where now the mi are rational numbers. The algebraic integers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> are then precisely those elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> where the mi are all integers. Working <a href="/wiki/Local_ring" title="Local ring">locally</a> and using tools such as the <a href="/wiki/Frobenius_map" class="mw-redirect" title="Frobenius map">Frobenius map</a>, it is always possible to explicitly compute such a basis, and it is now standard for <a href="/wiki/Computer_algebra_system" title="Computer algebra system">computer algebra systems</a> to have built-in programs to do this. <div class="mw-heading mw-heading3"><h3 id="Power_basis">Power basis</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=11" title="Edit section: Power basis">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> be a number field of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">. Among all possible bases of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> (seen as a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">-vector space), there are particular ones known as <a href="/wiki/Power_basis" class="mw-redirect" title="Power basis">power bases</a>, that are bases of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{x}=\{1,x,x^{2},\ldots ,x^{n-1}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{x}=\{1,x,x^{2},\ldots ,x^{n-1}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74e59376954b3417c57b6b4712b06e0db14e386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.131ex; height:3.176ex;" alt="{\displaystyle B_{x}=\{1,x,x^{2},\ldots ,x^{n-1}\}}"></dd></dl> for some element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9075ef2df706bf2764f76182a1ddf55ff231a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle x\in K}">. By the <a href="/wiki/Primitive_element_theorem" title="Primitive element theorem">primitive element theorem</a>, there exists such an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">, called a <a href="/wiki/Primitive_element_(field_theory)" class="mw-redirect" title="Primitive element (field theory)">primitive element</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> can be chosen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdf444010bc455aa47531544d6e7128fc2e9483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.936ex; height:2.509ex;" alt="{\displaystyle B_{x}}"> is a basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> as a free Z-module, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{x}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdf444010bc455aa47531544d6e7128fc2e9483" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.936ex; height:2.509ex;" alt="{\displaystyle B_{x}}"> is called a <a href="/wiki/Power_integral_basis" class="mw-redirect" title="Power integral basis">power integral basis</a>, and the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is called a <a href="/wiki/Monogenic_field" title="Monogenic field">monogenic field</a>. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial<a href="#cite_note-3">[3]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-x^{2}-2x-8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-x^{2}-2x-8.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/227da19005c988929a9ae626af0c592ff8af25a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.59ex; height:2.843ex;" alt="{\displaystyle x^{3}-x^{2}-2x-8.}"> <div class="mw-heading mw-heading2"><h2 id="Regular_representation,_trace_and_discriminant">Regular representation, trace and discriminant</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=12" title="Edit section: Regular representation, trace and discriminant">edit</a>]</div> Recall that any field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6782a2935122dafec2947d1d7ba168be4399ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }"> has a unique <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">-vector space structure. Using the multiplication in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> over the base field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> may be represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> <a href="/wiki/Matrix_(math)" class="mw-redirect" title="Matrix (math)">matrices</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A(x)=(a_{ij})_{1\leq i,j\leq n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=A(x)=(a_{ij})_{1\leq i,j\leq n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ada59d68e41306a242a6327572a9f3a0c2cd98e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.638ex; height:3.009ex;" alt="{\displaystyle A=A(x)=(a_{ij})_{1\leq i,j\leq n}}"> by requiring <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xe_{i}=\sum _{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xe_{i}=\sum _{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in \mathbb {Q} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/733710e526ce9837ef5f385b6868b8af6c932b98" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.113ex; height:7.176ex;" alt="{\displaystyle xe_{i}=\sum _{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in \mathbb {Q} .}"> Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\ldots ,e_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c60c38b7e2450d62e9dc496b89f8e5c96c77cecf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}"> is a fixed basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, viewed as a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">-vector space. The rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebea6cd2813c330c798921a2894b358f7b643917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}"> are uniquely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and the choice of a basis since any element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> can be uniquely represented as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of the basis elements. This way of associating a matrix to any element of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is called the <a href="/wiki/Regular_representation" title="Regular representation">regular representation</a>. The square matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> represents the effect of multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> in the given basis. It follows that if the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is represented by a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, then the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"> is represented by the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BA}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8efb97e621ab9b49f8498a49704690bdeb2698" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle BA}">. <a href="/wiki/Invariant_(mathematics)" title="Invariant (mathematics)">Invariants</a> of matrices, such as the <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a>, <a href="/wiki/Determinant" title="Determinant">determinant</a>, and <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a>, depend solely on the field element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and not on the basis. In particular, the trace of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078651e6d1a522e8955b73059fbd63e13aec616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle A(x)}"> is called the <a href="/wiki/Field_trace" title="Field trace">trace</a> of the field element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tr}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Tr}}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c54cc3c4f501cb7805342d417bccbaf80f843d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.729ex; height:2.843ex;" alt="{\displaystyle {\text{Tr}}(x)}">, and the determinant is called the <a href="/wiki/Field_norm" title="Field norm">norm</a> of x and denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca1ee5a5ef6497108b2f6d7f1fc8a224a52062fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle N(x)}">. Now this can be generalized slightly by instead considering a field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb79e236fa3e355b2c2e37b37ccf84ad3f9dabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle K/L}"> and giving an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">-basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. Then, there is an associated matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{K/L}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{K/L}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0cdc9d7f902758a4f081febba6a6851118b863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.516ex; height:3.176ex;" alt="{\displaystyle A_{K/L}(x)}">, which has trace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tr}}_{K/L}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Tr}}_{K/L}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d5d51655880df64ca8a82f69bf750f539fea28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.363ex; height:3.176ex;" alt="{\displaystyle {\text{Tr}}_{K/L}(x)}"> and norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{N}}_{K/L}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>N</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{N}}_{K/L}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce5e755f28cc9804592dadec48cb52d415e0e23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.516ex; height:3.176ex;" alt="{\displaystyle {\text{N}}_{K/L}(x)}"> defined as the trace and determinant of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{K/L}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{K/L}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0cdc9d7f902758a4f081febba6a6851118b863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.516ex; height:3.176ex;" alt="{\displaystyle A_{K/L}(x)}">. <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=13" title="Edit section: Example">edit</a>]</div> Consider the field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1fd6a15e30edf0b6adff4d5a9275a94238e2dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.708ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\theta )}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\zeta _{3}{\sqrt[{3}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\zeta _{3}{\sqrt[{3}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a6ee9e10065b633def711f392ebc85ae9635079" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.36ex; height:3.009ex;" alt="{\displaystyle \theta =\zeta _{3}{\sqrt[{3}]{2}}}">. Then, we have a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">-basis given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,\zeta _{3}{\sqrt[{3}]{2}},\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>,</mo> <msubsup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,\zeta _{3}{\sqrt[{3}]{2}},\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6501115d758cb5a30d65e76a4dfe1d47d6f9ba5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.439ex; height:3.843ex;" alt="{\displaystyle \{1,\zeta _{3}{\sqrt[{3}]{2}},\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}\}}"> since any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Q} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Q} (\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ae5ebe3d9a0c4384f586cff9bd6eb448f9473a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.878ex; height:2.843ex;" alt="{\displaystyle x\in \mathbb {Q} (\theta )}"> can be expressed as some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">-linear combination <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b\zeta _{3}{\sqrt[{3}]{2}}+c\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}=a+b\theta +c\theta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>+</mo> <mi>c</mi> <msubsup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> </mrow> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>θ</mi> <mo>+</mo> <mi>c</mi> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b\zeta _{3}{\sqrt[{3}]{2}}+c\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}=a+b\theta +c\theta ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f47152325e64119d31924a03981dc491947a43aa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.048ex; height:3.843ex;" alt="{\displaystyle a+b\zeta _{3}{\sqrt[{3}]{2}}+c\zeta _{3}^{2}{\sqrt[{3}]{2^{2}}}=a+b\theta +c\theta ^{2}}"> Then, we can take some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in \mathbb {Q} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in \mathbb {Q} (\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34c3085d59ac17406fa3b183c28c6c390b086c1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.704ex; height:2.843ex;" alt="{\displaystyle y\in \mathbb {Q} (\theta )}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=y_{0}+y_{1}\theta +y_{2}\theta ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>θ</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=y_{0}+y_{1}\theta +y_{2}\theta ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72a66901f90f93edf1741ea5085f164bf38bf6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.75ex; height:3.009ex;" alt="{\displaystyle y=y_{0}+y_{1}\theta +y_{2}\theta ^{2}}"> and compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13939e6cddd7ba416fd805830d8f5d815c9b4e76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.164ex; height:2.009ex;" alt="{\displaystyle x\cdot y}">. Writing this out gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a(y_{0}+y_{1}\theta +y_{2}\theta ^{2})+\\b(2y_{2}+y_{0}\theta +y_{1}\theta ^{2})+\\c(2y_{1}+2y_{2}\theta +y_{0}\theta ^{2})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>a</mi> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>θ</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>θ</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo stretchy="false">(</mo> <mn>2</mn> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>θ</mi> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a(y_{0}+y_{1}\theta +y_{2}\theta ^{2})+\\b(2y_{2}+y_{0}\theta +y_{1}\theta ^{2})+\\c(2y_{1}+2y_{2}\theta +y_{0}\theta ^{2})\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02a26deb37db96095c0baeb13087acedc785bb11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.072ex; margin-bottom: -0.266ex; width:22.025ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}a(y_{0}+y_{1}\theta +y_{2}\theta ^{2})+\\b(2y_{2}+y_{0}\theta +y_{1}\theta ^{2})+\\c(2y_{1}+2y_{2}\theta +y_{0}\theta ^{2})\end{aligned}}}"> We can find the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078651e6d1a522e8955b73059fbd63e13aec616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle A(x)}"> by writing out the associated matrix equation giving <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}{\begin{bmatrix}y_{0}\\y_{1}\\y_{2}\end{bmatrix}}={\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\by_{0}+ay_{1}+2cy_{2}\\cy_{0}+by_{1}+ay_{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>c</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>c</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>b</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}{\begin{bmatrix}y_{0}\\y_{1}\\y_{2}\end{bmatrix}}={\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\by_{0}+ay_{1}+2cy_{2}\\cy_{0}+by_{1}+ay_{2}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de31a691e4234da686cc1ad80323a3a23ac2d68b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:48.632ex; height:9.509ex;" alt="{\displaystyle {\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{bmatrix}}{\begin{bmatrix}y_{0}\\y_{1}\\y_{2}\end{bmatrix}}={\begin{bmatrix}ay_{0}+2cy_{1}+2by_{2}\\by_{0}+ay_{1}+2cy_{2}\\cy_{0}+by_{1}+ay_{2}\end{bmatrix}}}"> showing <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x)={\begin{bmatrix}a&2c&2b\\b&a&2c\\c&b&a\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mn>2</mn> <mi>c</mi> </mtd> <mtd> <mn>2</mn> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> <mtd> <mn>2</mn> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)={\begin{bmatrix}a&2c&2b\\b&a&2c\\c&b&a\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd05492a14a8acfacdd23257c86845d364cfa55e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.046ex; height:9.176ex;" alt="{\displaystyle A(x)={\begin{bmatrix}a&2c&2b\\b&a&2c\\c&b&a\end{bmatrix}}}"> We can then compute the trace and determinant with relative ease, giving the trace and norm. <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=14" title="Edit section: Properties">edit</a>]</div> By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr(x) is a <a href="/wiki/Linear_function" title="Linear function">linear function</a> of x, as expressed by Tr(x + y) = Tr(x) + Tr(y), Tr(λx) = λ Tr(x), and the norm is a multiplicative <a href="/wiki/Homogeneous_function" title="Homogeneous function">homogeneous function</a> of degree n: N(xy) = N(x) N(y), N(λx) = λn N(x). Here λ is a rational number, and x, y are any two elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. The <a href="/wiki/Trace_form" class="mw-redirect" title="Trace form">trace form</a> derived is a <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a> defined by means of the trace, as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tr_{K/L}:K\otimes _{L}K\to L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo>:</mo> <mi>K</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>K</mi> <mo stretchy="false">→</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tr_{K/L}:K\otimes _{L}K\to L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf31a9d34c5fb82b283ccc59cf2a907d14eac16" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.777ex; height:3.009ex;" alt="{\displaystyle Tr_{K/L}:K\otimes _{L}K\to L}"> by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Tr_{K/L}(x\otimes y)=Tr_{K/L}(x\cdot y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>T</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Tr_{K/L}(x\otimes y)=Tr_{K/L}(x\cdot y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd8ed4b188992675a72a463d217c96639041344" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:28.845ex; height:3.176ex;" alt="{\displaystyle Tr_{K/L}(x\otimes y)=Tr_{K/L}(x\cdot y)}"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Tr}}_{K/L}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Tr}}_{K/L}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d5d51655880df64ca8a82f69bf750f539fea28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.363ex; height:3.176ex;" alt="{\displaystyle {\text{Tr}}_{K/L}(x)}">. The integral trace form, an integer-valued <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a> is defined as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{ij}={\text{Tr}}_{K/\mathbb {Q} }(b_{i}b_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>Tr</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{ij}={\text{Tr}}_{K/\mathbb {Q} }(b_{i}b_{j})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c64d295da822abe9931aa02ea212d443eb4c059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:17.312ex; height:3.176ex;" alt="{\displaystyle t_{ij}={\text{Tr}}_{K/\mathbb {Q} }(b_{i}b_{j})}">, where b1, ..., bn is an integral basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. The <a href="/wiki/Discriminant_of_an_algebraic_number_field" title="Discriminant of an algebraic number field">discriminant</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is defined as det(t). It is an integer, and is an invariant property of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, not depending on the choice of integral basis. The matrix associated to an element x of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> can also be used to give other, equivalent descriptions of algebraic integers. An element x of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is an algebraic integer if and only if the characteristic polynomial pA of the matrix A associated to x is a monic polynomial with integer coefficients. Suppose that the matrix A that represents an element x has integer entries in some basis e. By the <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a>, pA(A) = 0, and it follows that pA(x) = 0, so that x is an algebraic integer. Conversely, if x is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> that is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix A. In this case it can be proven that A is an <a href="/wiki/Integer_matrix" title="Integer matrix">integer matrix</a> in a suitable basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. The property of being an algebraic integer is defined in a way that is independent of a choice of a basis in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. <div class="mw-heading mw-heading3"><h3 id="Example_with_integral_basis">Example with integral basis</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=15" title="Edit section: Example with integral basis">edit</a>]</div> Consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4742935ac0cef90f6880c81a80f8969f72c38838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.112ex; height:2.843ex;" alt="{\displaystyle K=\mathbb {Q} (x)}">, where x satisfies x3 − 11x2 + x + 1 = 0. Then an integral basis is [1, x, 1/2(x2 + 1)], and the corresponding integral trace form is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}3&11&61\\11&119&653\\61&653&3589\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>11</mn> </mtd> <mtd> <mn>61</mn> </mtd> </mtr> <mtr> <mtd> <mn>11</mn> </mtd> <mtd> <mn>119</mn> </mtd> <mtd> <mn>653</mn> </mtd> </mtr> <mtr> <mtd> <mn>61</mn> </mtd> <mtd> <mn>653</mn> </mtd> <mtd> <mn>3589</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}3&11&61\\11&119&653\\61&653&3589\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8851abfe825a8582d53c877c55ce8801316910b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:19.606ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}3&11&61\\11&119&653\\61&653&3589\end{bmatrix}}.}"> The "3" in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. This basis element induces the identity map on the 3-dimensional vector space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. The trace of the matrix of the identity map on a 3-dimensional vector space is 3. The determinant of this is 1304 = 23·163, the field discriminant; in comparison the <a href="/wiki/Discriminant" title="Discriminant">root discriminant</a>, or discriminant of the polynomial, is 5216 = 25·163. <div class="mw-heading mw-heading2"><h2 id="Places">Places</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=16" title="Edit section: Places">edit</a>]</div> Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> This situation changed with the discovery of <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> by <a href="/wiki/Kurt_Hensel" title="Kurt Hensel">Hensel</a> in 1897; and now it is standard to consider all of the various possible embeddings of a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> into its various topological <a href="/wiki/Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">completions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ef71423f0df0aacb0b459a435df75f029200a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.027ex; height:2.843ex;" alt="{\displaystyle K_{\mathfrak {p}}}"> at once. A <a href="/wiki/Place_(mathematics)" class="mw-redirect" title="Place (mathematics)">place</a> of a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is an equivalence class of <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">absolute values</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"><a href="#cite_note-:0-6">[6]</a>pg 9. Essentially, an absolute value is a notion to measure the size of elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). The equivalence relation between absolute values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{0}\sim |\cdot |_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∼</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{0}\sim |\cdot |_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1df11201cec3b52b0ed42c6ebc4faa53528b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.153ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{0}\sim |\cdot |_{1}}"> is given by some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda \in \mathbb {R} _{>0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>></mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \in \mathbb {R} _{>0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f67340a17ff96735f04cb9f6833eef16127488d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.207ex; height:2.509ex;" alt="{\displaystyle \lambda \in \mathbb {R} _{>0}}"> such that<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{0}=|\cdot |_{1}^{\lambda }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{0}=|\cdot |_{1}^{\lambda }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8326c19d8d11e77be738f884d1310fae41b54a9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.289ex; height:3.509ex;" alt="{\displaystyle |\cdot |_{0}=|\cdot |_{1}^{\lambda }}">meaning we take the value of the norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a77d0f6359cf0e4825de67e8a7e9863dcc8410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle |\cdot |_{1}}"> to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">-th power. In general, the types of places fall into three regimes. Firstly (and mostly irrelevant), the trivial absolute value | |0, which takes the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> on all non-zero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9075ef2df706bf2764f76182a1ddf55ff231a92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.236ex; height:2.176ex;" alt="{\displaystyle x\in K}">. The second and third classes are Archimedean places and non-Archimedean (or ultrametric) places. The completion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> with respect to a place <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12da79bc40f5db475ca48fb532d349d2d9a4e67d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.027ex; height:3.343ex;" alt="{\displaystyle |\cdot |_{\mathfrak {p}}}"> is given in both cases by taking <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy sequences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">and dividing out <a href="/wiki/Null_sequence" class="mw-redirect" title="Null sequence">null sequences</a>, that is, sequences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810ff85512c8477ba6aff18f4480ffeeac0bb3e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.374ex; height:2.843ex;" alt="{\displaystyle \{x_{n}\}_{n\in \mathbb {N} }}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |x_{n}|_{\mathfrak {p}}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |x_{n}|_{\mathfrak {p}}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724899d37e4ba7c40886ac428f5b8902ec27f3c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.673ex; height:3.343ex;" alt="{\displaystyle |x_{n}|_{\mathfrak {p}}\to 0}">tends to zero when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> tends to infinity. This can be shown to be a field again, the so-called completion of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> at the given place <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12da79bc40f5db475ca48fb532d349d2d9a4e67d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:4.027ex; height:3.343ex;" alt="{\displaystyle |\cdot |_{\mathfrak {p}}}">, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ef71423f0df0aacb0b459a435df75f029200a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.027ex; height:2.843ex;" alt="{\displaystyle K_{\mathfrak {p}}}">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0854290031d92bf519f3ed79754edbe3eec1db77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.973ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} }">, the following non-trivial norms occur (<a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a>): the (usual) <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a>, sometimes denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cdfb936407ce40d20152b514afb6f4eb5a055e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.848ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{\infty }}">, which gives rise to the complete <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological field</a> of the real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">. On the other hand, for any prime number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">, the <a href="/wiki/P-adic_number" title="P-adic number">p-adic</a> absolute value is defined by <dl><dd>|q|p = p−n, where q = pn a/b and a and b are integers not divisible by p.</dd></dl> It is used to construct the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">. In contrast to the usual absolute value, the p-adic absolute value gets smaller when q is multiplied by p, leading to quite different behavior of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> as compared to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">. Note the general situation typically considered is taking a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> and considering a <a href="/wiki/Prime_ideal_of_a_valuation" class="mw-redirect" title="Prime ideal of a valuation">prime ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da65e1a5261a55b04d68f2fe2b5b77f290cc263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.005ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}"> for its associated <a href="/wiki/Algebraic_number" title="Algebraic number">ring of algebraic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. Then, there will be a unique place <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\mathfrak {p}}:K\to \mathbb {R} _{\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\mathfrak {p}}:K\to \mathbb {R} _{\geq 0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f3b344d3e17a5f6c82958e82ab4d13b641f05f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.655ex; height:3.343ex;" alt="{\displaystyle |\cdot |_{\mathfrak {p}}:K\to \mathbb {R} _{\geq 0}}"> called a non-Archimedean place. In addition, for every embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma :K\to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma :K\to \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/766fc641757adde3c09e48675bc44bbf3ef1f76c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.625ex; height:2.176ex;" alt="{\displaystyle \sigma :K\to \mathbb {C} }"> there will be a place called an Archimedean place, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\cdot |_{\sigma }:K\to \mathbb {R} _{\geq 0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>≥</mo> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\cdot |_{\sigma }:K\to \mathbb {R} _{\geq 0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b45762039fa76315516479aef13dbc2e3267a15b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.773ex; height:3.009ex;" alt="{\displaystyle |\cdot |_{\sigma }:K\to \mathbb {R} _{\geq 0}}">. This statement is a theorem also called <a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Examples_2">Examples</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=17" title="Edit section: Examples">edit</a>]</div> The field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} [x]/(x^{6}-2)=\mathbb {Q} (\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} [x]/(x^{6}-2)=\mathbb {Q} (\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45f0f36ce0574c51b55389ccfb6755c8649f2f8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.761ex; height:3.176ex;" alt="{\displaystyle K=\mathbb {Q} [x]/(x^{6}-2)=\mathbb {Q} (\theta )}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =\zeta {\sqrt[{6}]{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </mroot> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =\zeta {\sqrt[{6}]{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6c8d8053451aeb5549a4ba58a2374fe032195c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.382ex; height:3.009ex;" alt="{\displaystyle \theta =\zeta {\sqrt[{6}]{2}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c3916703cae7938143d38865f78f27faadd4ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }"> is a fixed 6th root of unity, provides a rich example for constructing explicit real and complex Archimedean embeddings, and non-Archimedean embeddings as well<a href="#cite_note-:0-6">[6]</a>pg 15-16. <div class="mw-heading mw-heading3"><h3 id="Archimedean_places">Archimedean places</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=18" title="Edit section: Archimedean places">edit</a>]</div> Here we use the standard notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"> for the number of real and complex embeddings used, respectively (see below). Calculating the archimedean places of a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is done as follows: let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> be a primitive element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, with minimal polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> (over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">). Over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> of factors of degree one are necessarily real, and replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> gives an embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">; the number of such embeddings is equal to the number of real roots of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. Restricting the standard absolute value on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> gives an archimedean absolute value on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">; such an absolute value is also referred to as a real place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. On the other hand, the roots of factors of degree two are pairs of <a href="/wiki/Complex_conjugate" title="Complex conjugate">conjugate</a> complex numbers, which allows for two conjugate embeddings into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">. Either one of this pair of embeddings can be used to define an absolute value on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">.<a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a> If all roots of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> above are real (respectively, complex) or, equivalently, any possible embedding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subseteq \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subseteq \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cee94e14cc805e769f6e87c872607afb7b67457b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.842ex; height:2.343ex;" alt="{\displaystyle K\subseteq \mathbb {C} }"> is actually forced to be inside <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> (resp. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }">), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is called <a href="/wiki/Totally_real_number_field" title="Totally real number field">totally real</a> (resp. <a href="/wiki/Totally_complex_number_field" class="mw-redirect" title="Totally complex number field">totally complex</a>).<a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> <div class="mw-heading mw-heading3"><h3 id="Non-Archimedean_or_ultrametric_places">Non-Archimedean or ultrametric places</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=19" title="Edit section: Non-Archimedean or ultrametric places">edit</a>]</div> To find the non-Archimedean places, let again <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> be as above. In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> splits in factors of various degrees, none of which are repeated, and the degrees of which add up to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, the degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">. For each of these <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adically irreducible factors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}">, we may suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"> and obtain an embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> into an algebraic extension of finite degree over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">. Such a <a href="/wiki/Local_field" title="Local field">local field</a> behaves in many ways like a number field, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">. By using this <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic norm map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{f_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{f_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960c3c79c0733772e676bc4f1ce9a98462a8a3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.529ex; height:2.843ex;" alt="{\displaystyle N_{f_{i}}}"> for the place <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}">, we may define an absolute value corresponding to a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adically irreducible factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> by<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |y|_{f_{i}}=|N_{f_{i}}(y)|_{p}^{1/m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |y|_{f_{i}}=|N_{f_{i}}(y)|_{p}^{1/m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b8a3be44117125afb5578c96c5c3e9b6b6fac3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.317ex; height:4.009ex;" alt="{\displaystyle |y|_{f_{i}}=|N_{f_{i}}(y)|_{p}^{1/m}}">Such an absolute value is called an <a href="/wiki/Ultrametric" class="mw-redirect" title="Ultrametric">ultrametric</a>, non-Archimedean or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">-adic place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. For any ultrametric place v we have that |x|v ≤ 1 for any x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">, since the minimal polynomial for x has integer factors, and hence its p-adic factorization has factors in Zp. Consequently, the norm term (constant term) for each factor is a p-adic integer, and one of these is the integer used for defining the absolute value for v. <div class="mw-heading mw-heading3"><h3 id="Prime_ideals_in_OK">Prime ideals in OK</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=20" title="Edit section: Prime ideals in OK">edit</a>]</div> For an ultrametric place v, the subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> defined by |x|v < 1 is an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. This relies on the ultrametricity of v: given x and y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">, then <dl><dd>|x + y|v ≤ max (|x|v, |y|v) < 1.</dd></dl> Actually, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> is even a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>. Conversely, given a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">, a <a href="/wiki/Discrete_valuation" title="Discrete valuation">discrete valuation</a> can be defined by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\mathfrak {p}}(x)=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathfrak {p}}(x)=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dd3635b6b47ed5368127ada8b75b86f5be20117" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.814ex; height:3.009ex;" alt="{\displaystyle v_{\mathfrak {p}}(x)=n}"> where n is the biggest integer such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in {\mathfrak {p}}^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in {\mathfrak {p}}^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed54950a8e669b84eabd54de7a07195ef9ffec3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.551ex; height:2.676ex;" alt="{\displaystyle x\in {\mathfrak {p}}^{n}}">, the n-fold power of the ideal. This valuation can be turned into an ultrametric place. Under this correspondence, (equivalence classes) of ultrametric places of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> correspond to prime ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K=\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0854290031d92bf519f3ed79754edbe3eec1db77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.973ex; height:2.509ex;" alt="{\displaystyle K=\mathbb {Q} }">, this gives back Ostrowski's theorem: any prime ideal in Z (which is necessarily by a single prime number) corresponds to a non-Archimedean place and vice versa. However, for more general number fields, the situation becomes more involved, as will be explained below. Yet another, equivalent way of describing ultrametric places is by means of <a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localizations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. Given an ultrametric place <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> on a number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, the corresponding localization is the subring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> such that | x |v ≤ 1. By the ultrametric property <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is a ring. Moreover, it contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. For every element x of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, at least one of x or x−1 is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Actually, since K×/T× can be shown to be isomorphic to the integers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is a <a href="/wiki/Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a>, in particular a <a href="/wiki/Local_ring" title="Local ring">local ring</a>. Actually, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is just the localization of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> at the prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T={\mathcal {O}}_{K,{\mathfrak {p}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\mathcal {O}}_{K,{\mathfrak {p}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9ec0d13ae8624f448fdc73cf216d130baab838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.557ex; height:2.843ex;" alt="{\displaystyle T={\mathcal {O}}_{K,{\mathfrak {p}}}}">. Conversely, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> is the maximal ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">. Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field. <div class="mw-heading mw-heading4"><h4 id="Lying_over_theorem_and_places">Lying over theorem and places</h4>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=21" title="Edit section: Lying over theorem and places">edit</a>]</div> Some of the basic theorems in algebraic number theory are the <a href="/wiki/Going_up_and_going_down" title="Going up and going down">going up and going down theorems</a>, which describe the behavior of some prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da65e1a5261a55b04d68f2fe2b5b77f290cc263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.005ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}"> when it is extended as an ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170fbb174bb98c93172809f50338f114bbd1163a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.201ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{L}}"> for some field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0381945929156997b99bb43e5b7067d18c9a84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}">. We say that an ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {o}}\subset {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {o}}\subset {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f581ceabd43e0ecdde7bd53ebafa52954782775f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.437ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {o}}\subset {\mathcal {O}}_{L}}"> lies over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {o}}\cap {\mathcal {O}}_{K}={\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mo>∩</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {o}}\cap {\mathcal {O}}_{K}={\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc08500a8d15efe67e62dd21b355779e7184ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.523ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {o}}\cap {\mathcal {O}}_{K}={\mathfrak {p}}}">. Then, one incarnation of the theorem states a prime ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e5abb6a4fe520baa5f8e9c6a6469b507cfe11f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.661ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})}"> lies over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">, hence there is always a surjective map<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})\to {\text{Spec}}({\mathcal {O}}_{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})\to {\text{Spec}}({\mathcal {O}}_{K})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e2e03dd9889d6231ab7b9f33c9db843c064ff21" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.277ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})\to {\text{Spec}}({\mathcal {O}}_{K})}">induced from the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}\hookrightarrow {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">↪</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}\hookrightarrow {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/359b3f94dc621f62987f8385de02b77b438b40ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.651ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}\hookrightarrow {\mathcal {O}}_{L}}">. Since there exists a correspondence between places and prime ideals, this means we can find places dividing a place that is induced from a field extension. That is, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is a place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, then there are places <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> that divide <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">, in the sense that their induced prime ideals divide the induced prime ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e5abb6a4fe520baa5f8e9c6a6469b507cfe11f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.661ex; height:2.843ex;" alt="{\displaystyle {\text{Spec}}({\mathcal {O}}_{L})}">. In fact, this observation is useful<a href="#cite_note-:0-6">[6]</a>pg 13 while looking at the base change of an algebraic field extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> to one of its completions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">. If we write<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K={\frac {\mathbb {Q} [X]}{Q(X)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K={\frac {\mathbb {Q} [X]}{Q(X)}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecfa38b8ed43b0a80fd8504cf46f676a3bc42a9d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:11.628ex; height:6.509ex;" alt="{\displaystyle K={\frac {\mathbb {Q} [X]}{Q(X)}}}">and write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> for the induced element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\in K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e5edb9aa479e98075e5aa684ac88272fd6e243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.887ex; height:2.176ex;" alt="{\displaystyle X\in K}">, we get a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/404428ca4a02fdc8551f895424b96b434cbdd64e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.284ex; height:2.843ex;" alt="{\displaystyle K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}}">. Explicitly, this decomposition is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&={\frac {\mathbb {Q} [X]}{Q(X)}}\otimes _{\mathbb {Q} }\mathbb {Q} _{p}\\&={\frac {\mathbb {Q} _{p}[X]}{Q(X)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&={\frac {\mathbb {Q} [X]}{Q(X)}}\otimes _{\mathbb {Q} }\mathbb {Q} _{p}\\&={\frac {\mathbb {Q} _{p}[X]}{Q(X)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8855f9403c8ed47bfe8070eceb2f1236df18a827" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:26.817ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&={\frac {\mathbb {Q} [X]}{Q(X)}}\otimes _{\mathbb {Q} }\mathbb {Q} _{p}\\&={\frac {\mathbb {Q} _{p}[X]}{Q(X)}}\end{aligned}}}">furthermore, the induced polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(X)\in \mathbb {Q} _{p}[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(X)\in \mathbb {Q} _{p}[X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5773e172bbebbff6a4775249758941eae96357d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.609ex; height:3.009ex;" alt="{\displaystyle Q(X)\in \mathbb {Q} _{p}[X]}"> decomposes as<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(X)=\prod _{v|p}Q_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </mrow> </munder> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(X)=\prod _{v|p}Q_{v}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38735c837efb8a86be84c7e545f5bde2fa66836" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:14.951ex; height:6.009ex;" alt="{\displaystyle Q(X)=\prod _{v|p}Q_{v}}">because of <a href="/wiki/Hensel%27s_lemma" title="Hensel's lemma">Hensel's lemma</a><a href="#cite_note-11">[11]</a>pg 129-131; hence<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&\cong {\frac {\mathbb {Q} _{p}[X]}{\prod _{v|p}Q_{v}(X)}}\\&\cong \bigoplus _{v|p}K_{v}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>K</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </mrow> </munder> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≅</mo> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>p</mi> </mrow> </munder> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&\cong {\frac {\mathbb {Q} _{p}[X]}{\prod _{v|p}Q_{v}(X)}}\\&\cong \bigoplus _{v|p}K_{v}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2706317a7d23540c47a1ed0e892eb970bf971a30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:25.522ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}K\otimes _{\mathbb {Q} }\mathbb {Q} _{p}&\cong {\frac {\mathbb {Q} _{p}[X]}{\prod _{v|p}Q_{v}(X)}}\\&\cong \bigoplus _{v|p}K_{v}\end{aligned}}}">Moreover, there are embeddings<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}i_{v}:&K\to K_{v}\\&\theta \mapsto \theta _{v}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>:</mo> </mtd> <mtd> <mi>K</mi> <mo stretchy="false">→</mo> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>θ</mi> <mo stretchy="false">↦</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}i_{v}:&K\to K_{v}\\&\theta \mapsto \theta _{v}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9696e110dfcc6e1e4d961813b605c5d85e5ace5a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.558ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}i_{v}:&K\to K_{v}\\&\theta \mapsto \theta _{v}\end{aligned}}}">where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{v}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660bba2634de6a1eb78a6d9e4f5c7acc8c406b91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.12ex; height:2.509ex;" alt="{\displaystyle \theta _{v}}"> is a root of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{v}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/138e2e2b18d08e567eda703007eac1cd6908eceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.868ex; height:2.509ex;" alt="{\displaystyle Q_{v}}"> giving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}=\mathbb {Q} _{p}(\theta _{v})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{v}=\mathbb {Q} _{p}(\theta _{v})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6908bb469ac5a5fa6a91a5ce3ab752dbf513239" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.898ex; height:3.009ex;" alt="{\displaystyle K_{v}=\mathbb {Q} _{p}(\theta _{v})}">; hence we could write<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{v}=i_{v}(K)\mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{v}=i_{v}(K)\mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7ac04e464eb59e004c9b80b569ced48551dd0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.676ex; height:3.009ex;" alt="{\displaystyle K_{v}=i_{v}(K)\mathbb {Q} _{p}}">as subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f9e7692267c8a29ed4d848c3421eee929c23c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.737ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} _{p}}"> (which is the completion of the algebraic closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">). <div class="mw-heading mw-heading2"><h2 id="Ramification">Ramification</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=22" title="Edit section: Ramification">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Schematic_depiction_of_ramification.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Schematic_depiction_of_ramification.svg/300px-Schematic_depiction_of_ramification.svg.png" decoding="async" width="300" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Schematic_depiction_of_ramification.svg/450px-Schematic_depiction_of_ramification.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Schematic_depiction_of_ramification.svg/600px-Schematic_depiction_of_ramification.svg.png 2x" data-file-width="573" data-file-height="197" /></a><figcaption>Schematic depiction of ramification: the fibers of almost all points in Y below consist of three points, except for two points in Y marked with dots, where the fibers consist of one and two points (marked in black), respectively. The map f is said to be ramified in these points of Y.</figcaption></figure> <a href="/wiki/Ramification_(mathematics)" title="Ramification (mathematics)">Ramification</a>, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> such that the <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimages</a> of all points y in Y consist only of finitely many points): the cardinality of the <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fibers</a> f−1(y) will generally have the same number of points, but it occurs that, in special points y, this number drops. For example, the map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \to \mathbb {C} ,z\mapsto z^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> <mi>z</mi> <mo stretchy="false">↦</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \to \mathbb {C} ,z\mapsto z^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9426aec56f28d20c2c4a1da826e9c6ca11a6e99c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.015ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} \to \mathbb {C} ,z\mapsto z^{n}}"></dd></dl> has n points in each fiber over t, namely the n (complex) roots of t, except in t = 0, where the fiber consists of only one element, z = 0. One says that the map is "ramified" in zero. This is an example of a <a href="/wiki/Branched_covering" title="Branched covering">branched covering</a> of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>. This intuition also serves to define <a href="/wiki/Splitting_of_prime_ideals_in_Galois_extensions" title="Splitting of prime ideals in Galois extensions">ramification in algebraic number theory</a>. Given a (necessarily finite) extension of number fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb79e236fa3e355b2c2e37b37ccf84ad3f9dabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle K/L}">, a prime ideal p of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170fbb174bb98c93172809f50338f114bbd1163a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.201ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{L}}"> generates the ideal pOK of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}">. This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by <dl><dd>pO<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> = q1e1 q2e2 ⋯ qmem</dd></dl> with uniquely determined prime ideals qi of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> and numbers (called ramification indices) ei. Whenever one ramification index is bigger than one, the prime p is said to ramify in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. The connection between this definition and the geometric situation is delivered by the map of <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectra</a> of rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Spec} {\mathcal {O}}_{K}\to \mathrm {Spec} {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">c</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Spec} {\mathcal {O}}_{K}\to \mathrm {Spec} {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/220c5b0346a10a6bbfaa676334841cd4bd064999" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.658ex; height:2.509ex;" alt="{\displaystyle \mathrm {Spec} {\mathcal {O}}_{K}\to \mathrm {Spec} {\mathcal {O}}_{L}}">. In fact, <a href="/wiki/Unramified_morphism" title="Unramified morphism">unramified morphisms</a> of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> are a direct generalization of unramified extensions of number fields. Ramification is a purely local property, i.e., depends only on the completions around the primes p and qi. The <a href="/wiki/Inertia_group" class="mw-redirect" title="Inertia group">inertia group</a> measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields. <div class="mw-heading mw-heading3"><h3 id="An_example">An example</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=23" title="Edit section: An example">edit</a>]</div> The following example illustrates the notions introduced above. In order to compute the ramification index of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bd4b9882e48d137f8e7cfde1b46af7b10df734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (x)}">, where <dl><dd>f(x) = x3 − x − 1 = 0,</dd></dl> at 23, it suffices to consider the field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{23}(x)/\mathbb {Q} _{23}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{23}(x)/\mathbb {Q} _{23}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bce54e73cc400ca8dfa1d7202dbca5a3e9f3bcf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.67ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{23}(x)/\mathbb {Q} _{23}}">. Up to 529 = 232 (i.e., <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> 529) f can be factored as <dl><dd>f(x) = (x + 181)(x2 − 181x − 38) = gh.</dd></dl> Substituting x = y + 10 in the first factor g modulo 529 yields y + 191, so the valuation | y |g for y given by g is | −191 |23 = 1. On the other hand, the same substitution in h yields y2 − 161y − 161 modulo 529. Since 161 = 7 × 23, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\vert y\right\vert _{h}={\sqrt {\left\vert 161\right\vert }}_{23}={\frac {1}{\sqrt {23}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow> <mo>|</mo> <mi>y</mi> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>|</mo> <mn>161</mn> <mo>|</mo> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>23</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\vert y\right\vert _{h}={\sqrt {\left\vert 161\right\vert }}_{23}={\frac {1}{\sqrt {23}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62ebac72f116628145dbf9f29221cd13d8feea75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.903ex; height:6.176ex;" alt="{\displaystyle \left\vert y\right\vert _{h}={\sqrt {\left\vert 161\right\vert }}_{23}={\frac {1}{\sqrt {23}}}}"></dd></dl> Since possible values for the absolute value of the place defined by the factor h are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two. The valuations of any element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> can be computed in this way using <a href="/wiki/Resultant" title="Resultant">resultants</a>. If, for example y = x2 − x − 1, using the resultant to eliminate x between this relationship and f = x3 − x − 1 = 0 gives y3 − 5y2 + 4y − 1 = 0. If instead we eliminate with respect to the factors g and h of f, we obtain the corresponding factors for the polynomial for y, and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of y for g and h (which are both 1 in this instance.) <div class="mw-heading mw-heading3"><h3 id="Dedekind_discriminant_theorem">Dedekind discriminant theorem</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=24" title="Edit section: Dedekind discriminant theorem">edit</a>]</div> Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"> where p divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime p divides the discriminant, then there is a p-place that ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60bd4b9882e48d137f8e7cfde1b46af7b10df734" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (x)}"> with x3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does. The other ramified place comes from the absolute value on the complex embedding of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. <div class="mw-heading mw-heading2"><h2 id="Galois_groups_and_Galois_cohomology">Galois groups and Galois cohomology</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=25" title="Edit section: Galois groups and Galois cohomology">edit</a>]</div> Generally in abstract algebra, field extensions K / L can be studied by examining the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> Gal(K / L), consisting of field automorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> leaving <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> elementwise fixed. As an example, the Galois group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54e54ce307c021ad21920923bcc1b3f8bfee53bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.268ex; height:2.843ex;" alt="{\displaystyle \mathrm {Gal} (\mathbb {Q} (\zeta _{n})/\mathbb {Q} )}"> of the cyclotomic field extension of degree n (see above) is given by (Z/nZ)×, the group of invertible elements in Z/nZ. This is the first stepstone into <a href="/wiki/Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>. In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension K / K of the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a>, leading to the <a href="/wiki/Absolute_Galois_group" title="Absolute Galois group">absolute Galois group</a> G := Gal(K / K) or just Gal(K), and to the extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6782a2935122dafec2947d1d7ba168be4399ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }">. The <a href="/wiki/Fundamental_theorem_of_Galois_theory" title="Fundamental theorem of Galois theory">fundamental theorem of Galois theory</a> links fields in between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> and its algebraic closure and closed subgroups of Gal(K). For example, the <a href="/wiki/Abelianization" class="mw-redirect" title="Abelianization">abelianization</a> (the biggest abelian quotient) Gab of G corresponds to a field referred to as the maximal <a href="/wiki/Abelian_extension" title="Abelian extension">abelian extension</a> Kab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By the <a href="/wiki/Kronecker%E2%80%93Weber_theorem" title="Kronecker–Weber theorem">Kronecker–Weber theorem</a>, the maximal abelian extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> is the extension generated by all <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>. For more general number fields, <a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>, specifically the <a href="/wiki/Artin_reciprocity_law" class="mw-redirect" title="Artin reciprocity law">Artin reciprocity law</a> gives an answer by describing Gab in terms of the <a href="/wiki/Idele_class_group" class="mw-redirect" title="Idele class group">idele class group</a>. Also notable is the <a href="/wiki/Hilbert_class_field" title="Hilbert class field">Hilbert class field</a>, the maximal abelian unramified field extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. It can be shown to be finite over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, its Galois group over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> is isomorphic to the class group of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, in particular its degree equals the class number h of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> (see above). In certain situations, the Galois group <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a> on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of <a href="/wiki/Group_cohomology" title="Group cohomology">group cohomology</a> for the Galois group Gal(K), also known as <a href="/wiki/Galois_cohomology" title="Galois cohomology">Galois cohomology</a>, which in the first place measures the failure of exactness of taking Gal(K)-invariants, but offers deeper insights (and questions) as well. For example, the Galois group G of a field extension L / K acts on L×, the nonzero elements of L. This Galois module plays a significant role in many arithmetic <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">dualities</a>, such as <a href="/wiki/Poitou-Tate_duality" class="mw-redirect" title="Poitou-Tate duality">Poitou-Tate duality</a>. The <a href="/wiki/Brauer_group" title="Brauer group">Brauer group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, originally conceived to classify <a href="/wiki/Division_algebra" title="Division algebra">division algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, can be recast as a cohomology group, namely H2(Gal (K, K×)). <div class="mw-heading mw-heading2"><h2 id="Local-global_principle">Local-global principle</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=26" title="Edit section: Local-global principle">edit</a>]</div> Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaves</a> reifies that idea in <a href="/wiki/Topology" title="Topology">topology</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>. <div class="mw-heading mw-heading3"><h3 id="Local_and_global_fields">Local and global fields</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=27" title="Edit section: Local and global fields">edit</a>]</div> Number fields share a great deal of similarity with another class of fields much used in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> known as <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function fields</a> of <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a> over <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. An example is Kp(T). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate rings</a> (the quotient fields of which are the function fields in question) of curves. Therefore, both types of field are called <a href="/wiki/Global_field" title="Global field">global fields</a>. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding <a href="/wiki/Local_field" title="Local field">local fields</a>. For number fields <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, the local fields are the completions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> at all places, including the archimedean ones (see <a href="/wiki/Local_analysis" title="Local analysis">local analysis</a>). For function fields, the local fields are completions of the local rings at all points of the curve for function fields. Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case. <div class="mw-heading mw-heading3"><h3 id="Hasse_principle">Hasse principle</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=28" title="Edit section: Hasse principle">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hasse_principle" title="Hasse principle">Hasse principle</a></div> A prototypical question, posed at a global level, is whether some polynomial equation has a solution in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. If this is the case, this solution is also a solution in all completions. The <a href="/wiki/Local-global_principle" class="mw-redirect" title="Local-global principle">local-global principle</a> or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, which is often easier, since analytic methods (classical analytic tools such as <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a> at the archimedean places and <a href="/wiki/P-adic_analysis" title="P-adic analysis">p-adic analysis</a> at the nonarchimedean places) can be used. This implication does not hold, however, for more general types of equations. However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where <a href="/wiki/Local_class_field_theory" title="Local class field theory">local class field theory</a> is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completions Kv can be explicitly determined, whereas the Galois groups of global fields, even of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"> are far less understood. <div class="mw-heading mw-heading3"><h3 id="Adeles_and_ideles">Adeles and ideles</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=29" title="Edit section: Adeles and ideles">edit</a>]</div> In order to assemble local data pertaining to all local fields attached to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">, the <a href="/wiki/Adele_ring" title="Adele ring">adele ring</a> is set up. A multiplicative variant is referred to as <a href="/wiki/Idele" class="mw-redirect" title="Idele">ideles</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=30" title="Edit section: See also">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Generalizations">Generalizations</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=31" title="Edit section: Generalizations">edit</a>]</div> <ul><li><a href="/wiki/Algebraic_function_field" title="Algebraic function field">Algebraic function field</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Algebraic_number_theory">Algebraic number theory</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=32" title="Edit section: Algebraic number theory">edit</a>]</div> <ul><li><a href="/wiki/Dirichlet%27s_unit_theorem" title="Dirichlet's unit theorem">Dirichlet's unit theorem</a>, <a href="/wiki/S-unit" title="S-unit">S-unit</a></li> <li><a href="/wiki/Kummer_extension" class="mw-redirect" title="Kummer extension">Kummer extension</a></li> <li><a href="/wiki/Minkowski%27s_theorem" title="Minkowski's theorem">Minkowski's theorem</a>, <a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">Geometry of numbers</a></li> <li><a href="/wiki/Chebotarev%27s_density_theorem" title="Chebotarev's density theorem">Chebotarev's density theorem</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Class_field_theory">Class field theory</h3>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=33" title="Edit section: Class field theory">edit</a>]</div> <ul><li><a href="/wiki/Ray_class_group" class="mw-redirect" title="Ray class group">Ray class group</a></li> <li><a href="/wiki/Decomposition_group" class="mw-redirect" title="Decomposition group">Decomposition group</a></li> <li><a href="/wiki/Genus_field" title="Genus field">Genus field</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=34" title="Edit section: Notes">edit</a>]</div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFIrelandRosen1998" class="citation cs2">Ireland, Kenneth; <a href="/wiki/Michael_Rosen_(mathematician)" title="Michael Rosen (mathematician)">Rosen, Michael</a> (1998), A Classical Introduction to Modern Number Theory, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97329-6" title="Special:BookSources/978-0-387-97329-6"><bdi>978-0-387-97329-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Classical+Introduction+to+Modern+Number+Theory&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-0-387-97329-6&rft.aulast=Ireland&rft.aufirst=Kenneth&rft.au=Rosen%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988">, Ch. 1.4 </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlochKato1990" class="citation cs2">Bloch, Spencer; <a href="/wiki/Kazuya_Kato" title="Kazuya Kato">Kato, Kazuya</a> (1990), "L-functions and Tamagawa numbers of motives", The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Boston, MA: Birkhäuser Boston, pp. 333–400, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1086888">1086888</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=L-functions+and+Tamagawa+numbers+of+motives&rft.btitle=The+Grothendieck+Festschrift%2C+Vol.+I&rft.place=Boston%2C+MA&rft.series=Progr.+Math.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E333-%3C%2Fspan%3E400&rft.pub=Birkh%C3%A4user+Boston&rft.date=1990&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1086888%23id-name%3DMR&rft.aulast=Bloch&rft.aufirst=Spencer&rft.au=Kato%2C+Kazuya&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFNarkiewicz2004">Narkiewicz 2004</a>, §2.2.6 </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1999" class="citation cs2">Kleiner, Israel (1999), "Field theory: from equations to axiomatization. I", <a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>, 106 (7): 677–684, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2589500">10.2307/2589500</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2589500">2589500</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1720431">1720431</a>, <q>To Dedekind, then, fields were subsets of the complex numbers.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Field+theory%3A+from+equations+to+axiomatization.+I&rft.volume=106&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E677-%3C%2Fspan%3E684&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1720431%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2589500%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2589500&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMac_Lane1981" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1981), "Mathematical models: a sketch for the philosophy of mathematics", The American Mathematical Monthly, 88 (7): 462–472, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2321751">10.2307/2321751</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2321751">2321751</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0628015">0628015</a>, <q>Empiricism sprang from the 19th-century view of mathematics as almost coterminal with theoretical physics.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Mathematical+models%3A+a+sketch+for+the+philosophy+of+mathematics&rft.volume=88&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E462-%3C%2Fspan%3E472&rft.date=1981&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D628015%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2321751%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2321751&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"> </li> <li id="cite_note-:0-6">^ <a href="#cite_ref-:0_6-0">a</a> <a href="#cite_ref-:0_6-1">b</a> <a href="#cite_ref-:0_6-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGras2003" class="citation book cs1">Gras, Georges (2003). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/883382066">Class field theory : from theory to practice</a>. Berlin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-11323-3" title="Special:BookSources/978-3-662-11323-3"><bdi>978-3-662-11323-3</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/883382066">883382066</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Class+field+theory+%3A+from+theory+to+practice&rft.place=Berlin&rft.date=2003&rft_id=info%3Aoclcnum%2F883382066&rft.isbn=978-3-662-11323-3&rft.aulast=Gras&rft.aufirst=Georges&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F883382066&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: location missing publisher (<a href="/wiki/Category:CS1_maint:_location_missing_publisher" title="Category:CS1 maint: location missing publisher">link</a>) </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> Cohn, Chapter 11 §C p. 108 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Conrad </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Cohn, Chapter 11 §C p. 108 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Conrad </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeukirch1999" class="citation book cs1">Neukirch, Jürgen (1999). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/851391469">Algebraic Number Theory</a>. Berlin, Heidelberg: Springer Berlin Heidelberg. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-03983-0" title="Special:BookSources/978-3-662-03983-0"><bdi>978-3-662-03983-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/851391469">851391469</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Number+Theory&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer+Berlin+Heidelberg&rft.date=1999&rft_id=info%3Aoclcnum%2F851391469&rft.isbn=978-3-662-03983-0&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F851391469&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Algebraic_number_field&action=edit&section=35" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeune2023" class="citation book cs1">Keune, Frans (2023). <a rel="nofollow" class="external text" href="https://books.radbouduniversitypress.nl/index.php/rup/catalog/book/number_fields">Number Fields</a>. Radboud University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789493296039" title="Special:BookSources/9789493296039"><bdi>9789493296039</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Fields&rft.pub=Radboud+University+Press&rft.date=2023&rft.isbn=9789493296039&rft.aulast=Keune&rft.aufirst=Frans&rft_id=https%3A%2F%2Fbooks.radbouduniversitypress.nl%2Findex.php%2Frup%2Fcatalog%2Fbook%2Fnumber_fields&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"></li> <li>Keith Conrad, <a rel="nofollow" class="external free" href="http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf">http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJanusz1996" class="citation cs2">Janusz, Gerald J. (1996), Algebraic Number Fields (2nd ed.), Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0429-2" title="Special:BookSources/978-0-8218-0429-2"><bdi>978-0-8218-0429-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Number+Fields&rft.place=Providence%2C+R.I.&rft.edition=2nd&rft.pub=American+Mathematical+Society&rft.date=1996&rft.isbn=978-0-8218-0429-2&rft.aulast=Janusz&rft.aufirst=Gerald+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"></li> <li>Helmut Hasse, Number Theory, Springer Classics in Mathematics Series (2002)</li> <li><a href="/wiki/Serge_Lang" title="Serge Lang">Serge Lang</a>, Algebraic Number Theory, second edition, Springer, 2000</li> <li>Richard A. Mollin, Algebraic Number Theory, CRC, 1999</li> <li>Ram Murty, Problems in Algebraic Number Theory, Second Edition, Springer, 2005</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNarkiewicz2004" class="citation cs2">Narkiewicz, Władysław (2004), Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics (3 ed.), Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-21902-6" title="Special:BookSources/978-3-540-21902-6"><bdi>978-3-540-21902-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2078267">2078267</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+and+analytic+theory+of+algebraic+numbers&rft.place=Berlin&rft.series=Springer+Monographs+in+Mathematics&rft.edition=3&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=978-3-540-21902-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2078267%23id-name%3DMR&rft.aulast=Narkiewicz&rft.aufirst=W%C5%82adys%C5%82aw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeukirch1999" class="citation cs2"><a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999), Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65399-8" title="Special:BookSources/978-3-540-65399-8"><bdi>978-3-540-65399-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+number+theory&rft.place=Berlin%2C+New+York&rft.series=Grundlehren+der+Mathematischen+Wissenschaften&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0956.11021%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1697859%23id-name%3DMR&rft.isbn=978-3-540-65399-8&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeukirchSchmidtWingberg2000" class="citation cs2"><a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a>; Schmidt, Alexander; Wingberg, Kay (2000), Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, vol. 323, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-66671-4" title="Special:BookSources/978-3-540-66671-4"><bdi>978-3-540-66671-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1737196">1737196</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1136.11001">1136.11001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cohomology+of+Number+Fields&rft.place=Berlin%2C+New+York&rft.series=Grundlehren+der+Mathematischen+Wissenschaften&rft.pub=Springer-Verlag&rft.date=2000&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1136.11001%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1737196%23id-name%3DMR&rft.isbn=978-3-540-66671-4&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rft.au=Schmidt%2C+Alexander&rft.au=Wingberg%2C+Kay&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+field" class="Z3988"></li> <li><a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a>, Basic Number Theory, third edition, Springer, 1995</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; 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Algebraic number field - Wikipedia