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class="vector-toc-numb">1.4</span> <span>Dirichlet</span> </div> </a> <ul id="toc-Dirichlet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dedekind" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dedekind"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Dedekind</span> </div> </a> <ul id="toc-Dedekind-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Hilbert</span> </div> </a> <ul id="toc-Hilbert-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Artin" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Artin"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>Artin</span> </div> </a> <ul id="toc-Artin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.8</span> <span>Modern theory</span> </div> </a> <ul id="toc-Modern_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_notions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_notions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Basic notions</span> </div> </a> <button aria-controls="toc-Basic_notions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Basic notions subsection</span> </button> <ul id="toc-Basic_notions-sublist" class="vector-toc-list"> <li id="toc-Failure_of_unique_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Failure_of_unique_factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Failure of unique factorization</span> </div> </a> <ul id="toc-Failure_of_unique_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Factorization_into_prime_ideals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Factorization_into_prime_ideals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Factorization into prime ideals</span> </div> </a> <ul id="toc-Factorization_into_prime_ideals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideal_class_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal_class_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Ideal class group</span> </div> </a> <ul id="toc-Ideal_class_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_and_complex_embeddings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_and_complex_embeddings"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Real and complex embeddings</span> </div> </a> <ul id="toc-Real_and_complex_embeddings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Places" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Places"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Places</span> </div> </a> <ul id="toc-Places-sublist" class="vector-toc-list"> <li id="toc-Places_at_infinity_geometrically" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Places_at_infinity_geometrically"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5.1</span> <span>Places at infinity geometrically</span> </div> </a> <ul id="toc-Places_at_infinity_geometrically-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Units</span> </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zeta_function" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zeta_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.7</span> <span>Zeta function</span> </div> </a> <ul id="toc-Zeta_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Local_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.8</span> <span>Local fields</span> </div> </a> <ul id="toc-Local_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Major_results" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Major_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Major results</span> </div> </a> <button aria-controls="toc-Major_results-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Major results subsection</span> </button> <ul id="toc-Major_results-sublist" class="vector-toc-list"> <li id="toc-Finiteness_of_the_class_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finiteness_of_the_class_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Finiteness of the class group</span> </div> </a> <ul id="toc-Finiteness_of_the_class_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dirichlet's_unit_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dirichlet's_unit_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Dirichlet's unit theorem</span> </div> </a> <ul id="toc-Dirichlet's_unit_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprocity_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reciprocity_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Reciprocity laws</span> </div> </a> <ul id="toc-Reciprocity_laws-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Class_number_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Class_number_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Class number formula</span> </div> </a> <ul id="toc-Class_number_formula-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_areas" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Related_areas"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related areas</span> </div> </a> <ul id="toc-Related_areas-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </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"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Introductory_texts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Introductory_texts"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Introductory texts</span> </div> </a> <ul id="toc-Introductory_texts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Intermediate_texts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Intermediate_texts"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Intermediate texts</span> </div> </a> <ul id="toc-Intermediate_texts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graduate_level_texts" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graduate_level_texts"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Graduate level texts</span> </div> </a> <ul id="toc-Graduate_level_texts-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Algebraic number theory</span></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 37 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-37" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">37 languages</span> </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/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%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="نظرية الأعداد الجبرية – Arabic" lang="ar" hreflang="ar" data-title="نظرية الأعداد الجبرية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%BAmberos_alxebraicos" title="Teoría de númberos alxebraicos – Asturian" lang="ast" hreflang="ast" data-title="Teoría de númberos alxebraicos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A4%E0%A6%A4%E0%A7%8D%E0%A6%A4%E0%A7%8D%E0%A6%AC" title="বীজগাণিতিক সংখ্যাতত্ত্ব – Bangla" lang="bn" hreflang="bn" data-title="বীজগাণিতিক সংখ্যাতত্ত্ব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba badge-Q70893996 mw-list-item" title=""><a href="https://ba.wikipedia.org/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D0%B8%D1%8F:%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%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Категория:Алгебраик һандар теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Категория:Алгебраик һандар теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D1%87%D0%B8%D1%81%D0%BB%D0%B0%D1%82%D0%B0" title="Алгебрична теория на числата – Bulgarian" lang="bg" hreflang="bg" data-title="Алгебрична теория на числата" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_nombres_algebraics" title="Teoria de nombres algebraics – Catalan" lang="ca" hreflang="ca" data-title="Teoria de nombres algebraics" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D0%B8%D1%81%D0%B5%D0%BF%D1%81%D0%B5%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%C4%83%D0%BB%D0%BB%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Хисепсен алгебрăлла теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Хисепсен алгебрăлла теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebraische_Zahlentheorie" title="Algebraische Zahlentheorie – German" lang="de" hreflang="de" data-title="Algebraische Zahlentheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B9%CE%BA%CE%AE_%CE%B8%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8E%CE%BD" title="Αλγεβρική θεωρία αριθμών – Greek" lang="el" hreflang="el" data-title="Αλγεβρική θεωρία αριθμών" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%BAmeros_algebraicos" title="Teoría de números algebraicos – Spanish" lang="es" hreflang="es" data-title="Teoría de números algebraicos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_aljebraikoen_teoria" title="Zenbaki aljebraikoen teoria – Basque" lang="eu" hreflang="eu" data-title="Zenbaki aljebraikoen teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%AC%D8%A8%D8%B1%DB%8C_%D8%A7%D8%B9%D8%AF%D8%A7%D8%AF" title="نظریه جبری اعداد – Persian" lang="fa" hreflang="fa" data-title="نظریه جبری اعداد" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_alg%C3%A9brique_des_nombres" title="Théorie algébrique des nombres – French" lang="fr" hreflang="fr" data-title="Théorie algébrique des nombres" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%BAmeros_alx%C3%A9bricos" title="Teoría de números alxébricos – Galician" lang="gl" hreflang="gl" data-title="Teoría de números alxébricos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></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%EB%A1%A0" title="대수적 수론 – Korean" lang="ko" hreflang="ko" data-title="대수적 수론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_bilangan_aljabar" title="Teori bilangan aljabar – Indonesian" lang="id" hreflang="id" data-title="Teori bilangan aljabar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_algebrica_dei_numeri" title="Teoria algebrica dei numeri – Italian" lang="it" hreflang="it" data-title="Teoria algebrica dei numeri" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="תורת המספרים האלגברית – Hebrew" lang="he" hreflang="he" data-title="תורת המספרים האלגברית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Algebrai_sz%C3%A1melm%C3%A9let" title="Algebrai számelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Algebrai számelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B5%80%E0%B4%9C%E0%B5%80%E0%B4%AF_%E0%B4%B8%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BE_%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82" title="ബീജീയ സംഖ്യാ ഗണിതം – Malayalam" lang="ml" hreflang="ml" data-title="ബീജീയ സംഖ്യാ ഗണിതം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_bilangan_algebra" title="Teori bilangan algebra – Malay" lang="ms" hreflang="ms" data-title="Teori bilangan algebra" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%80%E1%80%B9%E1%80%81%E1%80%9B%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%94%E1%80%8A%E1%80%BA%E1%80%B8%E1%80%80%E1%80%BB_%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9E%E1%80%AE%E1%80%A1%E1%80%AD%E1%80%AF%E1%80%9B%E1%80%AE" title="အက္ခရာသင်္ချာနည်းကျ ကိန်းသီအိုရီ – Burmese" lang="my" hreflang="my" data-title="အက္ခရာသင်္ချာနည်းကျ ကိန်းသီအိုရီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra%C3%AFsche_getaltheorie" title="Algebraïsche getaltheorie – Dutch" lang="nl" hreflang="nl" data-title="Algebraïsche getaltheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E7%9A%84%E6%95%B4%E6%95%B0%E8%AB%96" title="代数的整数論 – Japanese" lang="ja" hreflang="ja" data-title="代数的整数論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a 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hreflang="ru" data-title="Алгебраическая теория чисел" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Algebraic_number_theory" title="Algebraic number theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Algebraic number theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Algebarska_teorija_brojeva" title="Algebarska teorija brojeva – Serbian" lang="sr" hreflang="sr" data-title="Algebarska teorija brojeva" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link 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class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%97%D1%87%D0%BD%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D1%87%D0%B8%D1%81%D0%B5%D0%BB" title="Алгебраїчна теорія чисел – Ukrainian" lang="uk" hreflang="uk" data-title="Алгебраїчна теорія чисел" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_s%E1%BB%91_%C4%91%E1%BA%A1i_s%E1%BB%91" title="Lý thuyết số đại số – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết số đại số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E6%95%B0%E8%AE%BA" title="代数数论 – Wu" lang="wuu" hreflang="wuu" data-title="代数数论" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BB%A3%E6%95%B8%E6%95%B8%E8%AB%96" title="代數數論 – Cantonese" lang="yue" hreflang="yue" data-title="代數數論" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BB%A3%E6%95%B8%E6%95%B8%E8%AB%96" title="代數數論 – Chinese" lang="zh" hreflang="zh" data-title="代數數論" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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<div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of number theory</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul 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.sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 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.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;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory</span><br /><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;"><b><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a></b> <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> <p><b><a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a></b> </p> <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> <p><b><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a></b> </p> <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> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span></dd></dl></dd></dl> <p><b>Related structures</b> </p> <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;"><b><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a></b> <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> <p><b><a class="mw-selflink selflink">Algebraic number theory</a></b> </p> <dl><dd>• <a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo <span class="texhtml mvar" style="font-style:italic;">n</span></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"><i>p</i>-adic integers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}"></span></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer <i>p</i>-ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 })}"></span></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;"><b><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a></b> <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> <p><b><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></b> </p><p><b><a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a></b> </p><p><b><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></b> </p> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <b><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></b></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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Disqvisitiones-800.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/220px-Disqvisitiones-800.jpg" decoding="async" width="220" height="368" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/330px-Disqvisitiones-800.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Disqvisitiones-800.jpg/440px-Disqvisitiones-800.jpg 2x" data-file-width="478" data-file-height="800" /></a><figcaption>Title page of the first edition of <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a>, one of the founding works of modern algebraic number theory</figcaption></figure> <p><b>Algebraic number theory</b> is a branch of <a href="/wiki/Number_theory" title="Number theory">number theory</a> that uses the techniques of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> to study the <a href="/wiki/Integers" class="mw-redirect" title="Integers">integers</a>, <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a> and their <a href="/wiki/Rings_of_integers" class="mw-redirect" title="Rings of integers">rings of integers</a>, <a href="/wiki/Finite_field" title="Finite field">finite fields</a>, and <a href="/wiki/Algebraic_function_field" title="Algebraic function field">function fields</a>. These properties, such as whether a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> admits <a href="/wiki/Unique_factorization" class="mw-redirect" title="Unique factorization">unique factorization</a>, the behavior of <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a>, and the <a href="/wiki/Galois_group" title="Galois group">Galois groups</a> of <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, can resolve questions of primary importance in number theory, like the existence of solutions to <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equations</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Diophantus">Diophantus</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=2" title="Edit section: Diophantus"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The beginnings of algebraic number theory can be traced to Diophantine equations,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> named after the 3rd-century <a href="/wiki/Alexandria" title="Alexandria">Alexandrian</a> mathematician, <a href="/wiki/Diophantus" title="Diophantus">Diophantus</a>, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers <i>x</i> and <i>y</i> such that their sum, and the sum of their squares, equal two given numbers <i>A</i> and <i>B</i>, respectively: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=x+y\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=x+y\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dea23b05205ce0d6bb8f81a9914c25d10fcff0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.748ex; height:2.509ex;" alt="{\displaystyle A=x+y\ }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=x^{2}+y^{2}.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=x^{2}+y^{2}.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca02421f6448257da3066d993b9cec13bf5b979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.916ex; height:3.009ex;" alt="{\displaystyle B=x^{2}+y^{2}.\ }"></span></dd></dl> <p>Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation <br /> <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = <i>z</i><sup>2</sup> are given by the <a href="/wiki/Pythagorean_triple" title="Pythagorean triple">Pythagorean triples</a>, originally solved by the Babylonians (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 1800 BC</span>).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Solutions to linear Diophantine equations, such as 26<i>x</i> + 65<i>y</i> = 13, may be found using the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> (c. 5th century BC).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Diophantus's major work was the <i><a href="/wiki/Arithmetica" title="Arithmetica">Arithmetica</a></i>, of which only a portion has survived. </p> <div class="mw-heading mw-heading3"><h3 id="Fermat">Fermat</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=3" title="Edit section: Fermat"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> was first <a href="/wiki/Conjectured" class="mw-redirect" title="Conjectured">conjectured</a> by <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> in 1637, famously in the margin of a copy of <i>Arithmetica</i> where he claimed he had a proof that was too large to fit in the margin. No successful proof was published until 1995 despite the efforts of countless mathematicians during the 358 intervening years. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the <a href="/wiki/Modularity_theorem" title="Modularity theorem">modularity theorem</a> in the 20th century. </p> <div class="mw-heading mw-heading3"><h3 id="Gauss">Gauss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=4" title="Edit section: Gauss"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the founding works of algebraic number theory, the <i><b>Disquisitiones Arithmeticae</b></i> (<a href="/wiki/Latin" title="Latin">Latin</a>: <i>Arithmetical Investigations</i>) is a textbook of number theory written in Latin<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> by <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> in 1798 when Gauss was 21 and first published in 1801 when he was 24. In this book Gauss brings together results in number theory obtained by mathematicians such as Fermat, <a href="/wiki/Euler" class="mw-redirect" title="Euler">Euler</a>, <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a> and <a href="/wiki/Adrien-Marie_Legendre" title="Adrien-Marie Legendre">Legendre</a> and adds important new results of his own. Before the <i>Disquisitiones</i> was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways. </p><p>The <i>Disquisitiones</i> was the starting point for the work of other nineteenth century <a href="/wiki/Europe" title="Europe">European</a> mathematicians including <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a>, <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a> and <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>. Many of the annotations given by Gauss are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; we can now read them as containing the germs of the theories of <a href="/wiki/L-function" title="L-function">L-functions</a> and <a href="/wiki/Complex_multiplication" title="Complex multiplication">complex multiplication</a>, in particular. </p> <div class="mw-heading mw-heading3"><h3 id="Dirichlet">Dirichlet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=5" title="Edit section: Dirichlet"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a couple of papers in 1838 and 1839 <a href="/wiki/Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a> proved the first <a href="/wiki/Class_number_formula" title="Class number formula">class number formula</a>, for <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> (later refined by his student <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a>). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number fields</a>.<sup id="cite_ref-Elstrodt_5-0" class="reference"><a href="#cite_note-Elstrodt-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Based on his research of the structure of the <a href="/wiki/Unit_group" class="mw-redirect" title="Unit group">unit group</a> of <a href="/wiki/Quadratic_field" title="Quadratic field">quadratic fields</a>, he proved the <a href="/wiki/Dirichlet_unit_theorem" class="mw-redirect" title="Dirichlet unit theorem">Dirichlet unit theorem</a>, a fundamental result in algebraic number theory.<sup id="cite_ref-Kanemitsu_6-0" class="reference"><a href="#cite_note-Kanemitsu-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>He first used the <a href="/wiki/Pigeonhole_principle" title="Pigeonhole principle">pigeonhole principle</a>, a basic counting argument, in the proof of a theorem in <a href="/wiki/Diophantine_approximation" title="Diophantine approximation">diophantine approximation</a>, later named after him <a href="/wiki/Dirichlet%27s_approximation_theorem" title="Dirichlet's approximation theorem">Dirichlet's approximation theorem</a>. He published important contributions to Fermat's last theorem, for which he proved the cases <i>n</i> = 5 and <i>n</i> = 14, and to the <a href="/wiki/Quartic_reciprocity" title="Quartic reciprocity">biquadratic reciprocity law</a>.<sup id="cite_ref-Elstrodt_5-1" class="reference"><a href="#cite_note-Elstrodt-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Dirichlet_divisor_problem" class="mw-redirect" title="Dirichlet divisor problem">Dirichlet divisor problem</a>, for which he found the first results, is still an unsolved problem in number theory despite later contributions by other researchers. </p> <div class="mw-heading mw-heading3"><h3 id="Dedekind">Dedekind</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=6" title="Edit section: Dedekind"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>'s study of Lejeune Dirichlet's work was what led him to his later study of algebraic number fields and ideals. In 1863, he published Lejeune Dirichlet's lectures on number theory as <i><a href="/wiki/Vorlesungen_%C3%BCber_Zahlentheorie" title="Vorlesungen über Zahlentheorie">Vorlesungen über Zahlentheorie</a></i> ("Lectures on Number Theory") about which it has been written that: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>"Although the book is assuredly based on Dirichlet's lectures, and although Dedekind himself referred to the book throughout his life as Dirichlet's, the book itself was entirely written by Dedekind, for the most part after Dirichlet's death." (Edwards 1983)</p></blockquote> <p>1879 and 1894 editions of the <i>Vorlesungen</i> included supplements introducing the notion of an ideal, fundamental to <a href="/wiki/Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring theory</a>. (The word "Ring", introduced later by <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, does not appear in Dedekind's work.) Dedekind defined an ideal as a subset of a set of numbers, composed of <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a> that satisfy polynomial equations with integer coefficients. The concept underwent further development in the hands of Hilbert and, especially, of <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>. Ideals generalize Ernst Eduard Kummer's <a href="/wiki/Ideal_number" title="Ideal number">ideal numbers</a>, devised as part of Kummer's 1843 attempt to prove Fermat's Last Theorem. </p> <div class="mw-heading mw-heading3"><h3 id="Hilbert">Hilbert</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=7" title="Edit section: Hilbert"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> unified the field of algebraic number theory with his 1897 treatise <i><a href="/wiki/Zahlbericht" title="Zahlbericht">Zahlbericht</a></i> (literally "report on numbers"). He also resolved a significant number-theory <a href="/wiki/Waring%27s_problem" title="Waring's problem">problem formulated by Waring</a> in 1770. As with <a href="#The_finiteness_theorem">the finiteness theorem</a>, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> He then had little more to publish on the subject; but the emergence of <a href="/wiki/Hilbert_modular_form" title="Hilbert modular form">Hilbert modular forms</a> in the dissertation of a student means his name is further attached to a major area. </p><p>He made a series of conjectures on <a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>. The concepts were highly influential, and his own contribution lives on in the names of the <a href="/wiki/Hilbert_class_field" title="Hilbert class field">Hilbert class field</a> and of the <a href="/wiki/Hilbert_symbol" title="Hilbert symbol">Hilbert symbol</a> of <a href="/wiki/Local_class_field_theory" title="Local class field theory">local class field theory</a>. Results were mostly proved by 1930, after work by <a href="/wiki/Teiji_Takagi" title="Teiji Takagi">Teiji Takagi</a>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Artin">Artin</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=8" title="Edit section: Artin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a> established the <a href="/wiki/Artin_reciprocity_law" class="mw-redirect" title="Artin reciprocity law">Artin reciprocity law</a> in a series of papers (1924; 1927; 1930). This law is a general theorem in number theory that forms a central part of global class field theory.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The term "<a href="/wiki/Reciprocity_law_(mathematics)" class="mw-redirect" title="Reciprocity law (mathematics)">reciprocity law</a>" refers to a long line of more concrete number theoretic statements which it generalized, from the <a href="/wiki/Quadratic_reciprocity_law" class="mw-redirect" title="Quadratic reciprocity law">quadratic reciprocity law</a> and the reciprocity laws of <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Eisenstein</a> and Kummer to Hilbert's product formula for the <a href="/wiki/Hilbert_symbol" title="Hilbert symbol">norm symbol</a>. Artin's result provided a partial solution to <a href="/wiki/Hilbert%27s_ninth_problem" title="Hilbert's ninth problem">Hilbert's ninth problem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Modern_theory">Modern theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=9" title="Edit section: Modern theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Around 1955, Japanese mathematicians <a href="/wiki/Goro_Shimura" title="Goro Shimura">Goro Shimura</a> and <a href="/wiki/Yutaka_Taniyama" title="Yutaka Taniyama">Yutaka Taniyama</a> observed a possible link between two apparently completely distinct, branches of mathematics, <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a> and <a href="/wiki/Modular_form" title="Modular form">modular forms</a>. The resulting <a href="/wiki/Modularity_theorem" title="Modularity theorem">modularity theorem</a> (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is <a href="/wiki/Modular_elliptic_curve" title="Modular elliptic curve">modular</a>, meaning that it can be associated with a unique <a href="/wiki/Modular_form" title="Modular form">modular form</a>. </p><p>It was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a> found evidence supporting it, yet no proof; as a result the "astounding"<sup id="cite_ref-Singh_10-0" class="reference"><a href="#cite_note-Singh-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> conjecture was often known as the Taniyama–Shimura-Weil conjecture. It became a part of the <a href="/wiki/Langlands_program" title="Langlands program">Langlands program</a>, a list of important conjectures needing proof or disproof. </p><p>From 1993 to 1994, <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a> provided a proof of the <a href="/wiki/Modularity_theorem" title="Modularity theorem">modularity theorem</a> for <a href="/wiki/Semistable_elliptic_curve" class="mw-redirect" title="Semistable elliptic curve">semistable elliptic curves</a>, which, together with <a href="/wiki/Ribet%27s_theorem" title="Ribet's theorem">Ribet's theorem</a>, provided a proof for Fermat's Last Theorem. Almost every mathematician at the time had previously considered both Fermat's Last Theorem and the Modularity Theorem either impossible or virtually impossible to prove, even given the most cutting-edge developments. Wiles first announced his proof in June 1993<sup id="cite_ref-nyt_11-0" class="reference"><a href="#cite_note-nyt-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> in a version that was soon recognized as having a serious gap at a key point. The proof was corrected by Wiles, partly in collaboration with <a href="/wiki/Richard_Taylor_(mathematician)" title="Richard Taylor (mathematician)">Richard Taylor</a>, and the final, widely accepted version was released in September 1994, and formally published in 1995. The proof uses many techniques from <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> and <a href="/wiki/Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>, and other 20th-century techniques not available to Fermat. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_notions">Basic notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=10" title="Edit section: Basic notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Failure_of_unique_factorization">Failure of unique factorization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=11" title="Edit section: Failure of unique factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important property of the ring of integers is that it satisfies the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, that every (positive) integer has a factorization into a product of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, and this factorization is unique up to the ordering of the factors. This may no longer be true in the ring of integers <span class="texhtml"><i>O</i></span> of an algebraic number field <span class="texhtml"><i>K</i></span>. </p><p>A <i>prime element</i> is an element <span class="texhtml"><i>p</i></span> of <span class="texhtml"><i>O</i></span> such that if <span class="texhtml"><i>p</i></span> divides a product <span class="texhtml"><i>ab</i></span>, then it divides one of the factors <span class="texhtml"><i>a</i></span> or <span class="texhtml"><i>b</i></span>. This property is closely related to primality in the integers, because any positive integer satisfying this property is either <span class="texhtml">1</span> or a prime number. However, it is strictly weaker. For example, <span class="texhtml">−2</span> is not a prime number because it is negative, but it is a prime element. If factorizations into prime elements are permitted, then, even in the integers, there are alternative factorizations such as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6=2\cdot 3=(-2)\cdot (-3).}"> <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> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6=2\cdot 3=(-2)\cdot (-3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a506dbfd8f6557fed842ba79077579cb8a6f585a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.249ex; height:2.843ex;" alt="{\displaystyle 6=2\cdot 3=(-2)\cdot (-3).}"></span></dd></dl> <p>In general, if <span class="texhtml"><i>u</i></span> is a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>, meaning a number with a multiplicative inverse in <span class="texhtml"><i>O</i></span>, and if <span class="texhtml"><i>p</i></span> is a prime element, then <span class="texhtml"><i>up</i></span> is also a prime element. Numbers such as <span class="texhtml"><i>p</i></span> and <span class="texhtml"><i>up</i></span> are said to be <i>associate</i>. In the integers, the primes <span class="texhtml"><i>p</i></span> and <span class="texhtml">−<i>p</i></span> are associate, but only one of these is positive. Requiring that prime numbers be positive selects a unique element from among a set of associated prime elements. When <i>K</i> is not the rational numbers, however, there is no analog of positivity. For example, in the <a href="/wiki/Gaussian_integers" class="mw-redirect" title="Gaussian integers">Gaussian integers</a> <span class="texhtml"><b>Z</b>[<i>i</i>]</span>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> the numbers <span class="texhtml">1 + 2<i>i</i></span> and <span class="texhtml">−2 + <i>i</i></span> are associate because the latter is the product of the former by <span class="texhtml"><i>i</i></span>, but there is no way to single out one as being more canonical than the other. This leads to equations such as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f360ff0c643ca19aab141e73b825153eca20912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.79ex; height:2.843ex;" alt="{\displaystyle 5=(1+2i)(1-2i)=(2+i)(2-i),}"></span></dd></dl> <p>which prove that in <span class="texhtml"><b>Z</b>[<i>i</i>]</span>, it is not true that factorizations are unique up to the order of the factors. For this reason, one adopts the definition of unique factorization used in <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a> (UFDs). In a UFD, the prime elements occurring in a factorization are only expected to be unique up to units and their ordering. </p><p>However, even with this weaker definition, many rings of integers in algebraic number fields do not admit unique factorization. There is an algebraic obstruction called the ideal class group. When the ideal class group is trivial, the ring is a UFD. When it is not, there is a distinction between a prime element and an <a href="/wiki/Irreducible_element" title="Irreducible element">irreducible element</a>. An <i>irreducible element</i> <span class="texhtml"><i>x</i></span> is an element such that if <span class="texhtml"><i>x</i> = <i>yz</i></span>, then either <span class="texhtml"><i>y</i></span> or <span class="texhtml"><i>z</i></span> is a unit. These are the elements that cannot be factored any further. Every element in <i>O</i> admits a factorization into irreducible elements, but it may admit more than one. This is because, while all prime elements are irreducible, some irreducible elements may not be prime. For example, consider the ring <span class="texhtml"><b>Z</b>[√<span style="text-decoration:overline;">-5</span>]</span>.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In this ring, the numbers <span class="texhtml">3</span>, <span class="texhtml">2 + √<span style="text-decoration:overline;">-5</span></span> and <span class="texhtml">2 - √<span style="text-decoration:overline;">-5</span></span> are irreducible. This means that the number <span class="texhtml">9</span> has two factorizations into irreducible elements, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbbc22f71feb3e9ff84b4eb28f82346bd8a74af9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.66ex; height:3.176ex;" alt="{\displaystyle 9=3^{2}=(2+{\sqrt {-5}})(2-{\sqrt {-5}}).}"></span></dd></dl> <p>This equation shows that <span class="texhtml">3</span> divides the product <span class="texhtml">(2 + √<span style="text-decoration:overline;">-5</span>)(2 - √<span style="text-decoration:overline;">-5</span>) = 9</span>. If <span class="texhtml">3</span> were a prime element, then it would divide <span class="texhtml">2 + √<span style="text-decoration:overline;">-5</span></span> or <span class="texhtml">2 - √<span style="text-decoration:overline;">-5</span></span>, but it does not, because all elements divisible by <span class="texhtml">3</span> are of the form <span class="texhtml">3<i>a</i> + 3<i>b</i>√<span style="text-decoration:overline;">-5</span></span>. Similarly, <span class="texhtml">2 + √<span style="text-decoration:overline;">-5</span></span> and <span class="texhtml">2 - √<span style="text-decoration:overline;">-5</span></span> divide the product <span class="texhtml">3<sup>2</sup></span>, but neither of these elements divides <span class="texhtml">3</span> itself, so neither of them are prime. As there is no sense in which the elements <span class="texhtml">3</span>, <span class="texhtml">2 + √<span style="text-decoration:overline;">-5</span></span> and <span class="texhtml">2 - √<span style="text-decoration:overline;">-5</span></span> can be made equivalent, unique factorization fails in <span class="texhtml"><b>Z</b>[√<span style="text-decoration:overline;">-5</span>]</span>. Unlike the situation with units, where uniqueness could be repaired by weakening the definition, overcoming this failure requires a new perspective. </p> <div class="mw-heading mw-heading3"><h3 id="Factorization_into_prime_ideals">Factorization into prime ideals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=12" title="Edit section: Factorization into prime ideals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml"><i>I</i></span> is an ideal in <span class="texhtml"><i>O</i></span>, then there is always a factorization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msubsup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4594c7a141cd87d7196b07077c466f46df3898db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.214ex; height:3.176ex;" alt="{\displaystyle I={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}},}"></span></dd></dl> <p>where each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc456e8ce59c0b5f16401170ef68c996c25d512" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.962ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{i}}"></span> is a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>, and where this expression is unique up to the order of the factors. In particular, this is true if <span class="texhtml"><i>I</i></span> is the principal ideal generated by a single element. This is the strongest sense in which the ring of integers of a general number field admits unique factorization. In the language of ring theory, it says that rings of integers are <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domains</a>. </p><p>When <span class="texhtml"><i>O</i></span> is a UFD, every prime ideal is generated by a prime element. Otherwise, there are prime ideals which are not generated by prime elements. In <span class="texhtml"><b>Z</b>[√<span style="text-decoration:overline;">-5</span>]</span>, for instance, the ideal <span class="texhtml">(2, 1 + √<span style="text-decoration:overline;">-5</span>)</span> is a prime ideal which cannot be generated by a single element. </p><p>Historically, the idea of factoring ideals into prime ideals was preceded by Ernst Kummer's introduction of ideal numbers. These are numbers lying in an extension field <span class="texhtml"><i>E</i></span> of <span class="texhtml"><i>K</i></span>. This extension field is now known as the Hilbert class field. By the <a href="/wiki/Principal_ideal_theorem" title="Principal ideal theorem">principal ideal theorem</a>, every prime ideal of <span class="texhtml"><i>O</i></span> generates a principal ideal of the ring of integers of <span class="texhtml"><i>E</i></span>. A generator of this principal ideal is called an ideal number. Kummer used these as a substitute for the failure of unique factorization in <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic fields</a>. These eventually led Richard Dedekind to introduce a forerunner of ideals and to prove unique factorization of ideals. </p><p>An ideal which is prime in the ring of integers in one number field may fail to be prime when extended to a larger number field. Consider, for example, the prime numbers. The corresponding ideals <span class="texhtml"><i>p</i><b>Z</b></span> are prime ideals of the ring <span class="texhtml"><b>Z</b></span>. However, when this ideal is extended to the Gaussian integers to obtain <span class="texhtml"><i>p</i><b>Z</b>[<i>i</i>]</span>, it may or may not be prime. For example, the factorization <span class="texhtml">2 = (1 + <i>i</i>)(1 − <i>i</i>)</span> implies that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbf {Z} [i]=(1+i)\mathbf {Z} [i]\cdot (1-i)\mathbf {Z} [i]=((1+i)\mathbf {Z} [i])^{2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbf {Z} [i]=(1+i)\mathbf {Z} [i]\cdot (1-i)\mathbf {Z} [i]=((1+i)\mathbf {Z} [i])^{2};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a995ddb754419ff3e9fd05db0fd8d435e781c2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.313ex; height:3.176ex;" alt="{\displaystyle 2\mathbf {Z} [i]=(1+i)\mathbf {Z} [i]\cdot (1-i)\mathbf {Z} [i]=((1+i)\mathbf {Z} [i])^{2};}"></span></dd></dl> <p>note that because <span class="texhtml">1 + <i>i</i> = (1 − <i>i</i>) ⋅ <i>i</i></span>, the ideals generated by <span class="texhtml">1 + <i>i</i></span> and <span class="texhtml">1 − <i>i</i></span> are the same. A complete answer to the question of which ideals remain prime in the Gaussian integers is provided by <a href="/wiki/Fermat%27s_theorem_on_sums_of_two_squares" title="Fermat's theorem on sums of two squares">Fermat's theorem on sums of two squares</a>. It implies that for an odd prime number <span class="texhtml"><i>p</i></span>, <span class="texhtml"><i>p</i><b>Z</b>[<i>i</i>]</span> is a prime ideal if <span class="texhtml"><i>p</i> ≡ 3 (mod 4)</span> and is not a prime ideal if <span class="texhtml"><i>p</i> ≡ 1 (mod 4)</span>. This, together with the observation that the ideal <span class="texhtml">(1 + <i>i</i>)<b>Z</b>[<i>i</i>]</span> is prime, provides a complete description of the prime ideals in the Gaussian integers. Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory. Class field theory accomplishes this goal when <i>K</i> is an <a href="/wiki/Abelian_extension" title="Abelian extension">abelian extension</a> of <b>Q</b> (that is, a <a href="/wiki/Galois_extension" title="Galois extension">Galois extension</a> with <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> Galois group). </p> <div class="mw-heading mw-heading3"><h3 id="Ideal_class_group">Ideal class group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=13" title="Edit section: Ideal class group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unique factorization fails if and only if there are prime ideals that fail to be principal. The object which measures the failure of prime ideals to be principal is called the ideal class group. Defining the ideal class group requires enlarging the set of ideals in a ring of algebraic integers so that they admit a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> structure. This is done by generalizing ideals to <a href="/wiki/Fractional_ideal" title="Fractional ideal">fractional ideals</a>. A fractional ideal is an additive subgroup <span class="texhtml"><i>J</i></span> of <span class="texhtml"><i>K</i></span> which is closed under multiplication by elements of <span class="texhtml"><i>O</i></span>, meaning that <span class="texhtml"><i>xJ</i> ⊆ <i>J</i></span> if <span class="texhtml"><i>x</i> ∈ <i>O</i></span>. All ideals of <span class="texhtml"><i>O</i></span> are also fractional ideals. If <span class="texhtml"><i>I</i></span> and <span class="texhtml"><i>J</i></span> are fractional ideals, then the set <span class="texhtml"><i>IJ</i></span> of all products of an element in <span class="texhtml"><i>I</i></span> and an element in <span class="texhtml"><i>J</i></span> is also a fractional ideal. This operation makes the set of non-zero fractional ideals into a group. The group identity is the ideal <span class="texhtml">(1) = <i>O</i></span>, and the inverse of <span class="texhtml"><i>J</i></span> is a (generalized) <a href="/wiki/Ideal_quotient" title="Ideal quotient">ideal quotient</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{-1}=(O:J)=\{x\in K:xJ\subseteq O\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>O</mi> <mo>:</mo> <mi>J</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>K</mi> <mo>:</mo> <mi>x</mi> <mi>J</mi> <mo>⊆</mo> <mi>O</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{-1}=(O:J)=\{x\in K:xJ\subseteq O\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136a5ad5d497490c73ce74f4aa8329c733c7b6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.865ex; height:3.176ex;" alt="{\displaystyle J^{-1}=(O:J)=\{x\in K:xJ\subseteq O\}.}"></span></dd></dl> <p>The principal fractional ideals, meaning the ones of the form <span class="texhtml"><i>Ox</i></span> where <span class="texhtml"><i>x</i> ∈ <i>K</i><sup>×</sup></span>, form a subgroup of the group of all non-zero fractional ideals. The quotient of the group of non-zero fractional ideals by this subgroup is the ideal class group. Two fractional ideals <span class="texhtml"><i>I</i></span> and <span class="texhtml"><i>J</i></span> represent the same element of the ideal class group if and only if there exists an element <span class="texhtml"><i>x</i> ∈ <i>K</i></span> such that <span class="texhtml"><i>xI</i> = <i>J</i></span>. Therefore, the ideal class group makes two fractional ideals equivalent if one is as close to being principal as the other is. The ideal class group is generally denoted <span class="texhtml">Cl <i>K</i></span>, <span class="texhtml">Cl <i>O</i></span>, or <span class="texhtml">Pic <i>O</i></span> (with the last notation identifying it with the <a href="/wiki/Picard_group" title="Picard group">Picard group</a> in algebraic geometry). </p><p>The number of elements in the class group is called the <b>class number</b> of <i>K</i>. The class number of <span class="texhtml"><b>Q</b>(√<span style="text-decoration:overline;">-5</span>)</span> is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as <span class="texhtml">(2, 1 + √<span style="text-decoration:overline;">-5</span>)</span>. </p><p>The ideal class group has another description in terms of <a href="/wiki/Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisors</a>. These are formal objects which represent possible factorizations of numbers. The divisor group <span class="texhtml">Div <i>K</i></span> is defined to be the <a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian group</a> generated by the prime ideals of <span class="texhtml"><i>O</i></span>. There is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from <span class="texhtml"><i>K</i><sup>×</sup></span>, the non-zero elements of <span class="texhtml"><i>K</i></span> up to multiplication, to <span class="texhtml">Div <i>K</i></span>. Suppose that <span class="texhtml"><i>x</i> ∈ <i>K</i></span> satisfies </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f72ad479ff03698121cc83fbe4f7f6b01bdfa6c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.181ex; height:3.176ex;" alt="{\displaystyle (x)={\mathfrak {p}}_{1}^{e_{1}}\cdots {\mathfrak {p}}_{t}^{e_{t}}.}"></span></dd></dl> <p>Then <span class="texhtml">div <i>x</i></span> is defined to be the divisor </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {div} x=\sum _{i=1}^{t}e_{i}[{\mathfrak {p}}_{i}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>div</mi> <mo>⁡</mo> <mi>x</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </munderover> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {div} x=\sum _{i=1}^{t}e_{i}[{\mathfrak {p}}_{i}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78443e03990ae119341b0b6a9305b99eb6aabeab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.51ex; height:7.176ex;" alt="{\displaystyle \operatorname {div} x=\sum _{i=1}^{t}e_{i}[{\mathfrak {p}}_{i}].}"></span></dd></dl> <p>The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of <span class="texhtml">div</span> is the group of units in <span class="texhtml"><i>O</i></span>, while the <a href="/wiki/Cokernel" title="Cokernel">cokernel</a> is the ideal class group. In the language of <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, this says that there is an <a href="/wiki/Exact_sequence" title="Exact sequence">exact sequence</a> of abelian groups (written multiplicatively), </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\to O^{\times }\to K^{\times }{\xrightarrow {\text{div}}}\operatorname {Div} K\to \operatorname {Cl} K\to 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo stretchy="false">→</mo> <msup> <mi>O</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>→</mo> <mpadded width="+0.611em" lspace="0.278em" voffset=".15em"> <mtext>div</mtext> </mpadded> </mover> </mrow> <mi>Div</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mi>Cl</mi> <mo>⁡</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\to O^{\times }\to K^{\times }{\xrightarrow {\text{div}}}\operatorname {Div} K\to \operatorname {Cl} K\to 1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94e42629584629ddc3e1448204e5a4e15b8608ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.053ex; margin-top: -0.309ex; margin-bottom: -0.451ex; width:39.148ex; height:3.843ex;" alt="{\displaystyle 1\to O^{\times }\to K^{\times }{\xrightarrow {\text{div}}}\operatorname {Div} K\to \operatorname {Cl} K\to 1.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Real_and_complex_embeddings">Real and complex embeddings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=14" title="Edit section: Real and complex embeddings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some number fields, such as <span class="texhtml"><b>Q</b>(√<span style="text-decoration:overline;">2</span>)</span>, can be specified as subfields of the real numbers. Others, such as <span class="texhtml"><b>Q</b>(√<span style="text-decoration:overline;">−1</span>)</span>, cannot. Abstractly, such a specification corresponds to a field homomorphism <span class="texhtml"><i>K</i> → <b>R</b></span> or <span class="texhtml"><i>K</i> → <b>C</b></span>. These are called <b>real embeddings</b> and <b>complex embeddings</b>, respectively. </p><p>A real quadratic field <span class="texhtml"><b>Q</b>(√<span style="text-decoration:overline;"><i>a</i></span>)</span>, with <span class="texhtml"><i>a</i> ∈ <b>Q</b>, <i>a</i> > 0</span>, and <span class="texhtml"><i>a</i></span> not a <a href="/wiki/Square_number" title="Square number">perfect square</a>, is so-called because it admits two real embeddings but no complex embeddings. These are the field homomorphisms which send <span class="texhtml">√<span style="text-decoration:overline;"><i>a</i></span></span> to <span class="texhtml">√<span style="text-decoration:overline;"><i>a</i></span></span> and to <span class="texhtml">−√<span style="text-decoration:overline;"><i>a</i></span></span>, respectively. Dually, an imaginary quadratic field <span class="texhtml"><b>Q</b>(√<span style="text-decoration:overline;">−<i>a</i></span>)</span> admits no real embeddings but admits a conjugate pair of complex embeddings. One of these embeddings sends <span class="texhtml">√<span style="text-decoration:overline;">−<i>a</i></span></span> to <span class="texhtml">√<span style="text-decoration:overline;">−<i>a</i></span></span>, while the other sends it to its <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a>, <span class="texhtml">−√<span style="text-decoration:overline;">−<i>a</i></span></span>. </p><p>Conventionally, the number of real embeddings of <span class="texhtml"><i>K</i></span> is denoted <span class="texhtml"><i>r</i><sub>1</sub></span>, while the number of conjugate pairs of complex embeddings is denoted <span class="texhtml"><i>r</i><sub>2</sub></span>. The <b>signature</b> of <i>K</i> is the pair <span class="texhtml">(<i>r</i><sub>1</sub>, <i>r</i><sub>2</sub>)</span>. It is a theorem that <span class="texhtml"><i>r</i><sub>1</sub> + 2<i>r</i><sub>2</sub> = <i>d</i></span>, where <span class="texhtml"><i>d</i></span> is the degree of <span class="texhtml"><i>K</i></span>. </p><p>Considering all embeddings at once determines a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {C} ^{r_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⊕</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {C} ^{r_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83115880585d8cebf5436bedb0e93ddd6d3d9f4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.542ex; height:2.509ex;" alt="{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {C} ^{r_{2}}}"></span>, or equivalently <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {R} ^{2r_{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⊕</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {R} ^{2r_{2}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2f2906826d748f8f70f3d305ab080bf6e6cb00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.083ex; height:2.843ex;" alt="{\displaystyle M\colon K\to \mathbf {R} ^{r_{1}}\oplus \mathbf {R} ^{2r_{2}}.}"></span> This is called the <b>Minkowski embedding</b>. </p><p>The subspace of the codomain fixed by complex conjugation is a real vector space of dimension <span class="texhtml"><i>d</i></span> called <a href="/wiki/Minkowski_space_(number_field)" title="Minkowski space (number field)">Minkowski space</a>. Because the Minkowski embedding is defined by field homomorphisms, multiplication of elements of <span class="texhtml"><i>K</i></span> by an element <span class="texhtml"><i>x</i> ∈ <i>K</i></span> corresponds to multiplication by a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> in the Minkowski embedding. The <a href="/wiki/Dot_product" title="Dot product">dot product</a> on Minkowski space corresponds to the trace form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle x,y\rangle =\operatorname {Tr} (xy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>Tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\rangle =\operatorname {Tr} (xy)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55028092332125408da8ca75d4058d47356546d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.311ex; height:2.843ex;" alt="{\displaystyle \langle x,y\rangle =\operatorname {Tr} (xy)}"></span>. </p><p>The image of <span class="texhtml"><i>O</i></span> under the Minkowski embedding is a <span class="texhtml"><i>d</i></span>-dimensional <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a>. If <span class="texhtml"><i>B</i></span> is a basis for this lattice, then <span class="texhtml">det <i>B</i><sup>T</sup><i>B</i></span> is the <b>discriminant</b> of <span class="texhtml"><i>O</i></span>. The discriminant is denoted <span class="texhtml">Δ</span> or <span class="texhtml"><i>D</i></span>. The covolume of the image of <span class="texhtml"><i>O</i></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {|\Delta |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {|\Delta |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b40ce410ff06aa7b747b6e5704c7db9f78cd1c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.553ex; height:4.843ex;" alt="{\displaystyle {\sqrt {|\Delta |}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Places">Places</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=15" title="Edit section: Places"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Real and complex embeddings can be put on the same footing as prime ideals by adopting a perspective based on <a href="/wiki/Valuation_(algebra)" title="Valuation (algebra)">valuations</a>. Consider, for example, the integers. In addition to the usual <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> function |·| : <b>Q</b> → <b>R</b>, there are <a href="/wiki/P-adic_absolute_value" class="mw-redirect" title="P-adic absolute value">p-adic absolute value</a> functions |·|<sub>p</sub> : <b>Q</b> → <b>R</b>, defined for each prime number <i>p</i>, which measure divisibility by <i>p</i>. <a href="/wiki/Ostrowski%27s_theorem" title="Ostrowski's theorem">Ostrowski's theorem</a> states that these are all possible absolute value functions on <b>Q</b> (up to equivalence). Therefore, absolute values are a common language to describe both the real embedding of <b>Q</b> and the prime numbers. </p><p>A <b>place</b> of an algebraic number field is an equivalence class of <a href="/wiki/Absolute_value_(algebra)" title="Absolute value (algebra)">absolute value</a> functions on <i>K</i>. There are two types of places. There is a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}}"></span>-adic absolute value for each prime ideal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}}}"></span> of <i>O</i>, and, like the <i>p</i>-adic absolute values, it measures divisibility. These are called <b>finite places</b>. The other type of place is specified using a real or complex embedding of <i>K</i> and the standard absolute value function on <b>R</b> or <b>C</b>. These are <b>infinite places</b>. Because absolute values are unable to distinguish between a complex embedding and its conjugate, a complex embedding and its conjugate determine the same place. Therefore, there are <span class="texhtml"><i>r</i><sub>1</sub></span> real places and <span class="texhtml"><i>r</i><sub>2</sub></span> complex places. Because places encompass the primes, places are sometimes referred to as <b>primes</b>. When this is done, finite places are called <b>finite primes</b> and infinite places are called <b>infinite primes</b>. If <span class="texhtml"><i>v</i></span> is a valuation corresponding to an absolute value, then one frequently writes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\mid \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∣</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\mid \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77dd4c6fc35e8c9a32076d6ebe997745a8aa6ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.389ex; height:2.843ex;" alt="{\displaystyle v\mid \infty }"></span> to mean that <span class="texhtml"><i>v</i></span> is an infinite place and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\nmid \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>∤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v\nmid \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1f7d6fb89482044b4c147ca02ebdf6d891115b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.389ex; height:2.843ex;" alt="{\displaystyle v\nmid \infty }"></span> to mean that it is a finite place. </p><p>Considering all the places of the field together produces the <a href="/wiki/Adele_ring" title="Adele ring">adele ring</a> of the number field. The adele ring allows one to simultaneously track all the data available using absolute values. This produces significant advantages in situations where the behavior at one place can affect the behavior at other places, as in the <a href="/wiki/Artin_reciprocity_law" class="mw-redirect" title="Artin reciprocity law">Artin reciprocity law</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Places_at_infinity_geometrically">Places at infinity geometrically</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=16" title="Edit section: Places at infinity geometrically"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a geometric analogy for places at infinity which holds on the function fields of curves. For example, let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\mathbb {F} _{q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=\mathbb {F} _{q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d184911e91608307ebbe450dabc4d22dd92e84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.719ex; height:2.843ex;" alt="{\displaystyle k=\mathbb {F} _{q}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X/k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X/k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08f5158ac8b9c92451a3027ca030e159a4546ce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.354ex; height:2.843ex;" alt="{\displaystyle X/k}"></span> be a <a href="/wiki/Smooth_scheme" title="Smooth scheme">smooth</a>, <a href="/wiki/Projective_curve" class="mw-redirect" title="Projective curve">projective</a>, <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curve</a>. The <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=k(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=k(X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cd2d3659a2b48241b020200767252880adc4579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.84ex; height:2.843ex;" alt="{\displaystyle F=k(X)}"></span> has many absolute values, or places, and each corresponds to a point on the curve. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is the projective completion of an affine curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {X}}\subset \mathbb {A} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {X}}\subset \mathbb {A} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d0bd4f5c44e5c12906ca93d81a985aa605ef177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.975ex; height:2.843ex;" alt="{\displaystyle {\hat {X}}\subset \mathbb {A} ^{n}}"></span> then the points in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-{\hat {X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X-{\hat {X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8284d34bd02050798d9ca5d9340f14d3c5c3e70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.8ex; height:3.009ex;" alt="{\displaystyle X-{\hat {X}}}"></span> correspond to the places at infinity. Then, the completion of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> at one of these points gives an analogue of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>-adics. </p><p> For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {P} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\mathbb {P} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e76153dcdc3047dc450c0c23e0f1030ca57e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.553ex; height:2.676ex;" alt="{\displaystyle X=\mathbb {P} ^{1}}"></span> then its function field is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af82cc40c9f5016723dec48de1d786702bfc2c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.86ex; height:2.843ex;" alt="{\displaystyle k(t)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> is an indeterminant and the field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> is the field of fractions of polynomials in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. Then, a place <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/537e72b1a70774ae976de89f7919dc0e0a9bb86d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.187ex; height:2.343ex;" alt="{\displaystyle v_{p}}"></span> at a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae5d7ae3f7710fc989d0bd6dca038d74107f16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.08ex; height:2.509ex;" alt="{\displaystyle p\in X}"></span> measures the order of vanishing or the order of a pole of a fraction of polynomials <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)/q(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)/q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7825ceb4d6b20e32268676638416ad5fa8ba89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.769ex; height:2.843ex;" alt="{\displaystyle p(x)/q(x)}"></span> at the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=[2:1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>2</mn> <mo>:</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=[2:1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4307ceccfe0f808749a10601f98a74c01a50d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:9.913ex; height:2.843ex;" alt="{\displaystyle p=[2:1]}"></span>, so on the affine chart <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4a5419a8f07b0d246f0002346520867a0ff152a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.645ex; height:2.676ex;" alt="{\displaystyle x_{1}\neq 0}"></span> this corresponds to the point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\in \mathbb {A} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\in \mathbb {A} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/002b382d82ea77a353eaa5df7e472681e0ffb158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.735ex; height:2.676ex;" alt="{\displaystyle 2\in \mathbb {A} ^{1}}"></span>, the valuation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c423c2cb809c30cac725befa14ffbf4c85f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{2}}"></span> measures the <a href="/wiki/Order_of_vanishing" class="mw-redirect" title="Order of vanishing">order of vanishing</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}"></span> minus the order of vanishing of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38bbafe34a043d284f19231b946a76c0a4b16b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle q(x)}"></span> at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" 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 2}"></span>. The function field of the completion at the place <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb04c423c2cb809c30cac725befa14ffbf4c85f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{2}}"></span> is then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k((t-2))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k((t-2))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9fe75986fcd4be8a4ed0a378fc7ec938876139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.672ex; height:2.843ex;" alt="{\displaystyle k((t-2))}"></span> which is the field of <a href="/wiki/Power_series" title="Power series">power series</a> in the variable <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8808eaf3a51f810c1809d51795ddff7c8c842e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.842ex; height:2.343ex;" alt="{\displaystyle t-2}"></span>, so an element is of the form<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&a_{-k}(t-2)^{-k}+\cdots +a_{-1}(t-2)^{-1}+a_{0}+a_{1}(t-2)+a_{2}(t-2)^{2}+\cdots \\&=\sum _{n=-k}^{\infty }a_{n}(t-2)^{n}\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 /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&a_{-k}(t-2)^{-k}+\cdots +a_{-1}(t-2)^{-1}+a_{0}+a_{1}(t-2)+a_{2}(t-2)^{2}+\cdots \\&=\sum _{n=-k}^{\infty }a_{n}(t-2)^{n}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708458c3ac0a094a5e5a0c80ff9c5c99ea60bd6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:69.614ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}&a_{-k}(t-2)^{-k}+\cdots +a_{-1}(t-2)^{-1}+a_{0}+a_{1}(t-2)+a_{2}(t-2)^{2}+\cdots \\&=\sum _{n=-k}^{\infty }a_{n}(t-2)^{n}\end{aligned}}}"></span></p></blockquote><p>for some <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <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">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }"></span>. For the place at infinity, this corresponds to the function field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k((1/t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k((1/t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb1c5ef15d8e3b6ecb564b26467a3723213059c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.994ex; height:2.843ex;" alt="{\displaystyle k((1/t))}"></span> which are power series of the form<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"></p><blockquote class="templatequote"><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=-k}^{\infty }a_{n}(1/t)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n=-k}^{\infty }a_{n}(1/t)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c4028332eff82e9aaab3aeed493af219bd8849" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:13.427ex; height:7.009ex;" alt="{\displaystyle \sum _{n=-k}^{\infty }a_{n}(1/t)^{n}}"></span></p></blockquote> <div class="mw-heading mw-heading3"><h3 id="Units">Units</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=17" title="Edit section: Units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The integers have only two units, <span class="texhtml">1</span> and <span class="texhtml">−1</span>. Other rings of integers may admit more units. The Gaussian integers have four units, the previous two as well as <span class="texhtml">±<i>i</i></span>. The <a href="/wiki/Eisenstein_integers" class="mw-redirect" title="Eisenstein integers">Eisenstein integers</a> <span class="texhtml"><b>Z</b>[exp(2π<i>i</i> / 3)]</span> have six units. The integers in real quadratic number fields have infinitely many units. For example, in <span class="texhtml"><b>Z</b>[√<span style="text-decoration:overline;">3</span>]</span>, every power of <span class="texhtml">2 + √<span style="text-decoration:overline;">3</span></span> is a unit, and all these powers are distinct. </p><p>In general, the group of units of <span class="texhtml"><i>O</i></span>, denoted <span class="texhtml"><i>O</i><sup>×</sup></span>, is a finitely generated abelian group. The <a href="/wiki/Fundamental_theorem_of_finitely_generated_abelian_groups" class="mw-redirect" title="Fundamental theorem of finitely generated abelian groups">fundamental theorem of finitely generated abelian groups</a> therefore implies that it is a direct sum of a torsion part and a free part. Reinterpreting this in the context of a number field, the torsion part consists of the <a href="/wiki/Root_of_unity" title="Root of unity">roots of unity</a> that lie in <span class="texhtml"><i>O</i></span>. This group is cyclic. The free part is described by <a href="/wiki/Dirichlet%27s_unit_theorem" title="Dirichlet's unit theorem">Dirichlet's unit theorem</a>. This theorem says that rank of the free part is <span class="texhtml"><i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> − 1</span>. Thus, for example, the only fields for which the rank of the free part is zero are <span class="texhtml"><b>Q</b></span> and the imaginary quadratic fields. A more precise statement giving the structure of <i>O</i><sup>×</sup> ⊗<sub><b>Z</b></sub> <b>Q</b> as a <a href="/wiki/Galois_module" class="mw-redirect" title="Galois module">Galois module</a> for the Galois group of <i>K</i>/<b>Q</b> is also possible.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The free part of the unit group can be studied using the infinite places of <span class="texhtml"><i>K</i></span>. Consider the function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log |x|_{v})_{v}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>L</mi> <mo>:</mo> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mi>L</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log |x|_{v})_{v}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0610c67f2ca4d0111c466ff0697c047707b55ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.166ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}L:K^{\times }\to \mathbf {R} ^{r_{1}+r_{2}}\\L(x)=(\log |x|_{v})_{v}\end{cases}}}"></span></dd></dl> <p>where <span class="texhtml"><i>v</i></span> varies over the infinite places of <span class="texhtml"><i>K</i></span> and |·|<sub><i>v</i></sub> is the absolute value associated with <span class="texhtml"><i>v</i></span>. The function <span class="texhtml"><i>L</i></span> is a homomorphism from <span class="texhtml"><i>K</i><sup>×</sup></span> to a real vector space. It can be shown that the image of <span class="texhtml"><i>O</i><sup>×</sup></span> is a lattice that spans the hyperplane defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea7caa54d68b890655cf6a43bbdde7615a407464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.682ex; height:2.843ex;" alt="{\displaystyle x_{1}+\cdots +x_{r_{1}+r_{2}}=0.}"></span> The covolume of this lattice is the <b>regulator</b> of the number field. One of the simplifications made possible by working with the adele ring is that there is a single object, the <a href="/wiki/Idele_class_group" class="mw-redirect" title="Idele class group">idele class group</a>, that describes both the quotient by this lattice and the ideal class group. </p> <div class="mw-heading mw-heading3"><h3 id="Zeta_function">Zeta function</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=18" title="Edit section: Zeta function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Dedekind_zeta_function" title="Dedekind zeta function">Dedekind zeta function</a> of a number field, analogous to the <a href="/wiki/Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>, is an analytic object which describes the behavior of prime ideals in <span class="texhtml"><i>K</i></span>. When <span class="texhtml"><i>K</i></span> is an abelian extension of <span class="texhtml"><b>Q</b></span>, Dedekind zeta functions are products of <a href="/wiki/Dirichlet_L-function" title="Dirichlet L-function">Dirichlet L-functions</a>, with there being one factor for each <a href="/wiki/Dirichlet_character" title="Dirichlet character">Dirichlet character</a>. The trivial character corresponds to the Riemann zeta function. When <span class="texhtml"><i>K</i></span> is a <a href="/wiki/Galois_extension" title="Galois extension">Galois extension</a>, the Dedekind zeta function is the <a href="/wiki/Artin_L-function" title="Artin L-function">Artin L-function</a> of the <a href="/wiki/Regular_representation" title="Regular representation">regular representation</a> of the Galois group of <span class="texhtml"><i>K</i></span>, and it has a factorization in terms of irreducible <a href="/wiki/Artin_representation" class="mw-redirect" title="Artin representation">Artin representations</a> of the Galois group. </p><p>The zeta function is related to the other invariants described above by the <a href="/wiki/Class_number_formula" title="Class number formula">class number formula</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Local_fields">Local fields</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=19" title="Edit section: Local fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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 article: <a href="/wiki/Local_field" title="Local field">Local field</a></div> <p><a href="/wiki/Completion_(metric_space)" class="mw-redirect" title="Completion (metric space)">Completing</a> a number field <i>K</i> at a place <i>w</i> gives a <a href="/wiki/Complete_field" title="Complete field">complete field</a>. If the valuation is Archimedean, one obtains <b>R</b> or <b>C</b>, if it is non-Archimedean and lies over a prime <i>p</i> of the rationals, one obtains a finite extension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{w}/\mathbf {Q} _{p}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K_{w}/\mathbf {Q} _{p}:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07699ce2f2e149207322acce5ceb08fd5a7d1d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.904ex; height:3.176ex;" alt="{\displaystyle K_{w}/\mathbf {Q} _{p}:}"></span> a complete, discrete valued field with finite residue field. This process simplifies the arithmetic of the field and allows the local study of problems. For example, the <a href="/wiki/Kronecker%E2%80%93Weber_theorem" title="Kronecker–Weber theorem">Kronecker–Weber theorem</a> can be deduced easily from the analogous local statement. The philosophy behind the study of local fields is largely motivated by geometric methods. In algebraic geometry, it is common to study varieties locally at a point by localizing to a maximal ideal. Global information can then be recovered by gluing together local data. This spirit is adopted in algebraic number theory. Given a prime in the ring of algebraic integers in a number field, it is desirable to study the field locally at that prime. Therefore, one localizes the ring of algebraic integers to that prime and then completes the fraction field much in the spirit of geometry. </p> <div class="mw-heading mw-heading2"><h2 id="Major_results">Major results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=20" title="Edit section: Major results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Finiteness_of_the_class_group">Finiteness of the class group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=21" title="Edit section: Finiteness of the class group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the classical results in algebraic number theory is that the ideal class group of an algebraic number field <i>K</i> is finite. This is a consequence of <a href="/wiki/Minkowski%27s_bound" title="Minkowski's bound">Minkowski's theorem</a> since there are only finitely many <a href="/wiki/Integral_ideal" class="mw-redirect" title="Integral ideal">Integral ideals</a> with norm less than a fixed positive integer<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> <sup>page 78</sup>. The order of the class group is called the <a href="/wiki/Class_number_(number_theory)" class="mw-redirect" title="Class number (number theory)">class number</a>, and is often denoted by the letter <i>h</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Dirichlet's_unit_theorem"><span id="Dirichlet.27s_unit_theorem"></span>Dirichlet's unit theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=22" title="Edit section: Dirichlet's unit theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Dirichlet%27s_unit_theorem" title="Dirichlet's unit theorem">Dirichlet's unit theorem</a></div> <p>Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units <i>O</i><sup>×</sup> of the ring of integers <i>O</i>. Specifically, it states that <i>O</i><sup>×</sup> is isomorphic to <i>G</i> × <b>Z</b><sup><i>r</i></sup>, where <i>G</i> is the finite cyclic group consisting of all the roots of unity in <i>O</i>, and <i>r</i> = <i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> − 1 (where <i>r</i><sub>1</sub> (respectively, <i>r</i><sub>2</sub>) denotes the number of real embeddings (respectively, pairs of conjugate non-real embeddings) of <i>K</i>). In other words, <i>O</i><sup>×</sup> is a <a href="/wiki/Finitely_generated_abelian_group" title="Finitely generated abelian group">finitely generated abelian group</a> of <a href="/wiki/Rank_of_an_abelian_group" title="Rank of an abelian group">rank</a> <i>r</i><sub>1</sub> + <i>r</i><sub>2</sub> − 1 whose torsion consists of the roots of unity in <i>O</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Reciprocity_laws">Reciprocity laws</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=23" title="Edit section: Reciprocity laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Reciprocity_law" title="Reciprocity law">Reciprocity law</a></div> <p>In terms of the <a href="/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a>, the law of quadratic reciprocity for positive odd primes states </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/056c41d860587e6b676056d15ea25d3d054e3120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.366ex; height:6.176ex;" alt="{\displaystyle \left({\frac {p}{q}}\right)\left({\frac {q}{p}}\right)=(-1)^{{\frac {p-1}{2}}{\frac {q-1}{2}}}.}"></span></dd></dl> <p>A <b>reciprocity law</b> is a generalization of the <a href="/wiki/Law_of_quadratic_reciprocity" class="mw-redirect" title="Law of quadratic reciprocity">law of quadratic reciprocity</a>. </p><p>There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a <a href="/wiki/Power_residue_symbol" title="Power residue symbol">power residue symbol</a> (<i>p</i>/<i>q</i>) generalizing the <a href="/wiki/Legendre_symbol" title="Legendre symbol">quadratic reciprocity symbol</a>, that describes when a <a href="/wiki/Prime_number" title="Prime number">prime number</a> is an <i>n</i>th power residue <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> another prime, and gave a relation between (<i>p</i>/<i>q</i>) and (<i>q</i>/<i>p</i>). Hilbert reformulated the reciprocity laws as saying that a product over <i>p</i> of Hilbert symbols (<i>a</i>,<i>b</i>/<i>p</i>), taking values in roots of unity, is equal to 1. <a href="/wiki/Emil_Artin" title="Emil Artin">Artin</a>'s reformulated <a href="/wiki/Artin_reciprocity_law" class="mw-redirect" title="Artin reciprocity law">reciprocity law</a> states that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see. </p> <div class="mw-heading mw-heading3"><h3 id="Class_number_formula">Class number formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=24" title="Edit section: Class number formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Class_number_formula" title="Class number formula">Class number formula</a></div> <p>The <b>class number formula</b> relates many important invariants of a <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number field</a> to a special value of its Dedekind zeta function. </p> <div class="mw-heading mw-heading2"><h2 id="Related_areas">Related areas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=25" title="Edit section: Related areas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algebraic number theory interacts with many other mathematical disciplines. It uses tools from <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>. Via the analogy of function fields vs. number fields, it relies on techniques and ideas from algebraic geometry. Moreover, the study of higher-dimensional schemes over <b>Z</b> instead of number rings is referred to as <a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">arithmetic geometry</a>. Algebraic number theory is also used in the study of <a href="/wiki/Arithmetic_hyperbolic_3-manifold" title="Arithmetic hyperbolic 3-manifold">arithmetic hyperbolic 3-manifolds</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=26" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Class_field_theory" title="Class field theory">Class field theory</a></li> <li><a href="/wiki/Kummer_theory" title="Kummer theory">Kummer theory</a></li> <li><a href="/wiki/Locally_compact_field" title="Locally compact field">Locally compact field</a></li> <li><a href="/wiki/Tamagawa_number" title="Tamagawa number">Tamagawa number</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=27" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Stark, pp. 145–146.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Aczel, pp. 14–15.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Stark, pp. 44–47.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><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="CITEREFGaussWaterhouse2018" class="citation cs2">Gauss, Carl Friedrich; Waterhouse, William C. (2018) [1966], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DyFLDwAAQBAJ"><i>Disquisitiones Arithmeticae</i></a>, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4939-7560-0" title="Special:BookSources/978-1-4939-7560-0"><bdi>978-1-4939-7560-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Disquisitiones+Arithmeticae&rft.pub=Springer&rft.date=2018&rft.isbn=978-1-4939-7560-0&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rft.au=Waterhouse%2C+William+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDyFLDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-Elstrodt-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Elstrodt_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Elstrodt_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElstrodt2007" class="citation cs2">Elstrodt, Jürgen (2007), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20210522140235/https://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf">"The Life and Work of Gustav Lejeune Dirichlet (1805–1859)"</a> <span class="cs1-format">(PDF)</span>, <i>Clay Mathematics Proceedings</i>, archived from <a rel="nofollow" class="external text" href="http://www.uni-math.gwdg.de/tschinkel/gauss-dirichlet/elstrodt-new.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2021-05-22<span class="reference-accessdate">, retrieved <span class="nowrap">2007-12-25</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Clay+Mathematics+Proceedings&rft.atitle=The+Life+and+Work+of+Gustav+Lejeune+Dirichlet+%281805%E2%80%931859%29&rft.date=2007&rft.aulast=Elstrodt&rft.aufirst=J%C3%BCrgen&rft_id=http%3A%2F%2Fwww.uni-math.gwdg.de%2Ftschinkel%2Fgauss-dirichlet%2Felstrodt-new.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-Kanemitsu-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Kanemitsu_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKanemitsuChaohua_Jia2002" class="citation cs2">Kanemitsu, Shigeru; Chaohua Jia (2002), <i>Number theoretic methods: future trends</i>, Springer, pp. <span class="nowrap">271–</span>4, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-1080-4" title="Special:BookSources/978-1-4020-1080-4"><bdi>978-1-4020-1080-4</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+theoretic+methods%3A+future+trends&rft.pages=%3Cspan+class%3D%22nowrap%22%3E271-%3C%2Fspan%3E4&rft.pub=Springer&rft.date=2002&rft.isbn=978-1-4020-1080-4&rft.aulast=Kanemitsu&rft.aufirst=Shigeru&rft.au=Chaohua+Jia&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReid1996" class="citation cs2">Reid, Constance (1996), <i>Hilbert</i>, <a href="/wiki/Springer_Science_and_Business_Media" class="mw-redirect" title="Springer Science and Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94674-8" title="Special:BookSources/0-387-94674-8"><bdi>0-387-94674-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hilbert&rft.pub=Springer&rft.date=1996&rft.isbn=0-387-94674-8&rft.aulast=Reid&rft.aufirst=Constance&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">This work established Takagi as Japan's first mathematician of international stature.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHasse2010" class="citation cs2"><a href="/wiki/Helmut_Hasse" title="Helmut Hasse">Hasse, Helmut</a> (2010) [1967], "History of Class Field Theory", in <a href="/wiki/J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J. W. S.</a>; <a href="/wiki/Albrecht_Fr%C3%B6hlich" title="Albrecht Fröhlich">Fröhlich, Albrecht</a> (eds.), <i>Algebraic number theory</i> (2nd ed.), London: 9780950273426, pp. <span class="nowrap">266–</span>279, <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=0215665">0215665</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=History+of+Class+Field+Theory&rft.btitle=Algebraic+number+theory&rft.place=London&rft.pages=%3Cspan+class%3D%22nowrap%22%3E266-%3C%2Fspan%3E279&rft.edition=2nd&rft.pub=9780950273426&rft.date=2010&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0215665%23id-name%3DMR&rft.aulast=Hasse&rft.aufirst=Helmut&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-Singh-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Singh_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSingh1997" class="citation cs2"><a href="/wiki/Simon_Singh" title="Simon Singh">Singh, Simon</a> (1997), <i><a href="/wiki/Fermat%27s_Last_Theorem_(book)" title="Fermat's Last Theorem (book)">Fermat's Last Theorem</a></i>, Fourth Estate, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-85702-521-0" title="Special:BookSources/1-85702-521-0"><bdi>1-85702-521-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fermat%27s+Last+Theorem&rft.pub=Fourth+Estate&rft.date=1997&rft.isbn=1-85702-521-0&rft.aulast=Singh&rft.aufirst=Simon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-nyt-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-nyt_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKolata1993" class="citation news cs1">Kolata, Gina (24 June 1993). <a rel="nofollow" class="external text" href="https://www.nytimes.com/1993/06/24/us/at-last-shout-of-eureka-in-age-old-math-mystery.html">"At Last, Shout of 'Eureka!' In Age-Old Math Mystery"</a>. <i>The New York Times</i><span class="reference-accessdate">. Retrieved <span class="nowrap">21 January</span> 2013</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+New+York+Times&rft.atitle=At+Last%2C+Shout+of+%27Eureka%21%27+In+Age-Old+Math+Mystery&rft.date=1993-06-24&rft.aulast=Kolata&rft.aufirst=Gina&rft_id=https%3A%2F%2Fwww.nytimes.com%2F1993%2F06%2F24%2Fus%2Fat-last-shout-of-eureka-in-age-old-math-mystery.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">This notation indicates the ring obtained from <b>Z</b> by <a href="https://en.wiktionary.org/wiki/adjoin" class="extiw" title="wiktionary:adjoin">adjoining</a> to <b>Z</b> the element <i>i</i>.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">This notation indicates the ring obtained from <b>Z</b> by <a href="https://en.wiktionary.org/wiki/adjoin" class="extiw" title="wiktionary:adjoin">adjoining</a> to <b>Z</b> the element <span class="texhtml">√<span style="text-decoration:overline;">-5</span></span>.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">See proposition VIII.8.6.11 of <a href="#CITEREFNeukirchSchmidtWingberg2000">Neukirch, Schmidt & Wingberg 2000</a></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein" class="citation web cs1">Stein. <a rel="nofollow" class="external text" href="https://wstein.org/books/ant/ant.pdf">"A Computational Introduction to Algebraic Number Theory"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+Computational+Introduction+to+Algebraic+Number+Theory&rft.au=Stein&rft_id=https%3A%2F%2Fwstein.org%2Fbooks%2Fant%2Fant.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></span> </li> </ol></div></div> <ul><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), <i>Cohomology of Number Fields</i>, <i>Grundlehren der Mathematischen Wissenschaften</i>, vol. 323, Berlin: Springer-Verlag, <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:0948.11001">0948.11001</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&rft.series=%27%27Grundlehren+der+Mathematischen+Wissenschaften%27%27&rft.pub=Springer-Verlag&rft.date=2000&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0948.11001%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+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=28" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Introductory_texts">Introductory texts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=29" title="Edit section: Introductory texts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein2012" class="citation cs2">Stein, William (2012), <a rel="nofollow" class="external text" href="https://wstein.org/books/ant/ant.pdf"><i>Algebraic Number Theory, A Computational Approach</i></a> <span class="cs1-format">(PDF)</span></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%2C+A+Computational+Approach&rft.date=2012&rft.aulast=Stein&rft.aufirst=William&rft_id=https%3A%2F%2Fwstein.org%2Fbooks%2Fant%2Fant.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIrelandRosen2013" class="citation cs2">Ireland, Kenneth; Rosen, Michael (2013), <i>A classical introduction to modern number theory</i>, vol. 84, Springer, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-2103-4">10.1007/978-1-4757-2103-4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-2103-4" title="Special:BookSources/978-1-4757-2103-4"><bdi>978-1-4757-2103-4</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.pub=Springer&rft.date=2013&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-2103-4&rft.isbn=978-1-4757-2103-4&rft.aulast=Ireland&rft.aufirst=Kenneth&rft.au=Rosen%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStewartTall2015" class="citation cs2"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a>; <a href="/wiki/David_Tall" title="David Tall">Tall, David</a> (2015), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xXu9CgAAQBAJ"><i>Algebraic Number Theory and Fermat's Last Theorem</i></a>, CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4987-3840-8" title="Special:BookSources/978-1-4987-3840-8"><bdi>978-1-4987-3840-8</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+Theory+and+Fermat%27s+Last+Theorem&rft.pub=CRC+Press&rft.date=2015&rft.isbn=978-1-4987-3840-8&rft.aulast=Stewart&rft.aufirst=Ian&rft.au=Tall%2C+David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxXu9CgAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Intermediate_texts">Intermediate texts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=30" title="Edit section: Intermediate texts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMarcus2018" class="citation cs2">Marcus, Daniel A. (2018), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AjRjDwAAQBAJ"><i>Number Fields</i></a> (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-319-90233-3" title="Special:BookSources/978-3-319-90233-3"><bdi>978-3-319-90233-3</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.edition=2nd&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-319-90233-3&rft.aulast=Marcus&rft.aufirst=Daniel+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAjRjDwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Graduate_level_texts">Graduate level texts</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number_theory&action=edit&section=31" title="Edit section: Graduate level texts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCasselsFröhlich2010" class="citation cs2"><a href="/wiki/J._W._S._Cassels" title="J. W. S. Cassels">Cassels, J. W. S.</a>; <a href="/wiki/Albrecht_Fr%C3%B6hlich" title="Albrecht Fröhlich">Fröhlich, Albrecht</a>, eds. (2010) [1967], <i>Algebraic number theory</i> (2nd ed.), London: 9780950273426, <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=0215665">0215665</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=London&rft.edition=2nd&rft.pub=9780950273426&rft.date=2010&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0215665%23id-name%3DMR&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFröhlichTaylor1993" class="citation cs2"><a href="/wiki/Albrecht_Fr%C3%B6hlich" title="Albrecht Fröhlich">Fröhlich, Albrecht</a>; <a href="/wiki/Martin_J._Taylor" title="Martin J. Taylor">Taylor, Martin J.</a> (1993), <i>Algebraic number theory</i>, Cambridge Studies in Advanced Mathematics, vol. 27, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-43834-9" title="Special:BookSources/0-521-43834-9"><bdi>0-521-43834-9</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=1215934">1215934</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.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1993&rft.isbn=0-521-43834-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1215934%23id-name%3DMR&rft.aulast=Fr%C3%B6hlich&rft.aufirst=Albrecht&rft.au=Taylor%2C+Martin+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang1994" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (1994), <i>Algebraic number theory</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 110 (2 ed.), 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/978-0-387-94225-4" title="Special:BookSources/978-0-387-94225-4"><bdi>978-0-387-94225-4</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=1282723">1282723</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=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-0-387-94225-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1282723%23id-name%3DMR&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeukirch1999" class="citation book cs1"><a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999). <i>Algebraische Zahlentheorie</i>. <span title="German-language text"><i lang="de">Grundlehren der mathematischen Wissenschaften</i></span>. Vol. 322. Berlin: <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">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 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Number_theory52" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background:#ffb;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Number_theory" title="Template:Number theory"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_theory" title="Template talk:Number theory"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_theory" title="Special:EditPage/Template:Number theory"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_theory52" style="font-size:114%;margin:0 4em"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Algebraic number theory</a> (<a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>, <a href="/wiki/Non-abelian_class_field_theory" title="Non-abelian class field theory">non-abelian class field theory</a>, <a href="/wiki/Iwasawa_theory" title="Iwasawa theory">Iwasawa theory</a>, <a href="/wiki/Iwasawa%E2%80%93Tate_theory" class="mw-redirect" title="Iwasawa–Tate theory">Iwasawa–Tate theory</a>, <a href="/wiki/Kummer_theory" title="Kummer theory">Kummer theory</a>)</li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a> (<a href="/wiki/L-function" title="L-function">analytic theory of L-functions</a>, <a href="/wiki/Probabilistic_number_theory" title="Probabilistic number theory">probabilistic number theory</a>, <a href="/wiki/Sieve_theory" title="Sieve theory">sieve theory</a>)</li> <li><a href="/wiki/Geometry_of_numbers" title="Geometry of numbers">Geometric number theory</a></li> <li><a href="/wiki/Computational_number_theory" title="Computational number theory">Computational number theory</a></li> <li><a href="/wiki/Transcendental_number_theory" title="Transcendental number theory">Transcendental number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a> (<a href="/wiki/Arakelov_theory" title="Arakelov theory">Arakelov theory</a>, <a href="/wiki/Hodge%E2%80%93Arakelov_theory" title="Hodge–Arakelov theory">Hodge–Arakelov theory</a>)</li> <li><a href="/wiki/Arithmetic_combinatorics" title="Arithmetic combinatorics">Arithmetic combinatorics</a> (<a href="/wiki/Additive_number_theory" title="Additive number theory">additive number theory</a>)</li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic geometry</a> (<a href="/wiki/Anabelian_geometry" title="Anabelian geometry">anabelian geometry</a>, <a href="/wiki/P-adic_Hodge_theory" title="P-adic Hodge theory">p-adic Hodge theory</a>)</li> <li><a href="/wiki/Arithmetic_topology" title="Arithmetic topology">Arithmetic topology</a></li> <li><a href="/wiki/Arithmetic_dynamics" title="Arithmetic dynamics">Arithmetic dynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Number" title="Number">Numbers</a></li> <li><a href="/wiki/0" title="0">0</a></li> <li><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a></li> <li><a href="/wiki/1" title="1">Unity</a></li> <li><a href="/wiki/Prime_number" title="Prime number">Prime numbers</a></li> <li><a href="/wiki/Composite_number" title="Composite number">Composite numbers</a></li> <li><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a></li> <li><a href="/wiki/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Algebraic_number" title="Algebraic number">Algebraic numbers</a></li> <li><a href="/wiki/Transcendental_number" title="Transcendental number">Transcendental numbers</a></li> <li><a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> (<a href="/wiki/P-adic_analysis" title="P-adic analysis">p-adic analysis</a>)</li> <li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></li> <li><a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a></li> <li><a href="/wiki/Arithmetic_function" title="Arithmetic function">Arithmetic functions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#ffd;">Advanced concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Quadratic_form" title="Quadratic form">Quadratic forms</a></li> <li><a href="/wiki/Modular_form" title="Modular form">Modular forms</a></li> <li><a href="/wiki/L-function" title="L-function">L-functions</a></li> <li><a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equations</a></li> <li><a href="/wiki/Diophantine_approximation" title="Diophantine approximation">Diophantine approximation</a></li> <li><a href="/wiki/Irrationality_measure" title="Irrationality measure">Irrationality measure</a></li> <li><a href="/wiki/Simple_continued_fraction" title="Simple continued fraction">Simple continued fractions</a></li></ul> </div></td></tr><tr><td colspan="2" class="navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Number_theory" title="Category:Number theory">Category</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img 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