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<span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Degree of simple extensions of the rationals as a criterion to algebraicity</span> </div> </a> <ul id="toc-Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Field" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Field</span> </div> </a> <button aria-controls="toc-Field-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 Field subsection</span> </button> <ul id="toc-Field-sublist" class="vector-toc-list"> <li id="toc-Algebraic_closure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_closure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Algebraic closure</span> </div> </a> <ul id="toc-Algebraic_closure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Related_fields" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_fields"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related fields</span> </div> </a> <button aria-controls="toc-Related_fields-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 Related fields subsection</span> </button> <ul id="toc-Related_fields-sublist" class="vector-toc-list"> <li id="toc-Numbers_defined_by_radicals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numbers_defined_by_radicals"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Numbers defined by radicals</span> </div> </a> <ul id="toc-Numbers_defined_by_radicals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closed-form_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Closed-form_number"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Closed-form number</span> </div> </a> <ul id="toc-Closed-form_number-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebraic_integers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebraic_integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Algebraic integers</span> </div> </a> <ul id="toc-Algebraic_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_classes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Special_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Special classes</span> </div> </a> <ul id="toc-Special_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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</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 63 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-63" 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">63 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Algebraische_Zahl" title="Algebraische Zahl – Alemannic" lang="gsw" hreflang="gsw" data-title="Algebraische Zahl" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%AC%D8%A8%D8%B1%D9%8A" title="عدد جبري – Arabic" lang="ar" hreflang="ar" data-title="عدد جبري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</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" title="বীজগাণিতিক সংখ্যা – Bangla" lang="bn" hreflang="bn" data-title="বীজগাণিতিক সংখ্যা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D1%8B_%D0%BB%D1%96%D0%BA" title="Алгебраічны лік – Belarusian" lang="be" hreflang="be" data-title="Алгебраічны лік" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</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%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Алгебрично число – Bulgarian" lang="bg" hreflang="bg" data-title="Алгебрично число" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Algebarski_broj" title="Algebarski broj – Bosnian" lang="bs" hreflang="bs" data-title="Algebarski broj" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_algebraic" title="Nombre algebraic – Catalan" lang="ca" hreflang="ca" data-title="Nombre algebraic" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%C4%83%D0%BB%D0%BB%D0%B0_%D1%85%D0%B8%D1%81%D0%B5%D0%BF" title="Алгебрăлла хисеп – Chuvash" lang="cv" hreflang="cv" data-title="Алгебрăлла хисеп" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algebraick%C3%A9_%C4%8D%C3%ADslo" title="Algebraické číslo – Czech" lang="cs" hreflang="cs" data-title="Algebraické číslo" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Algebraiske_tal" title="Algebraiske tal – Danish" lang="da" hreflang="da" data-title="Algebraiske tal" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebraische_Zahl" title="Algebraische Zahl – German" lang="de" hreflang="de" data-title="Algebraische Zahl" 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%CF%8C%CF%82_%CE%B1%CF%81%CE%B9%CE%B8%CE%BC%CF%8C%CF%82" title="Αλγεβρικός αριθμός – Greek" lang="el" hreflang="el" data-title="Αλγεβρικός αριθμός" 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/N%C3%BAmero_algebraico" title="Número algebraico – Spanish" lang="es" hreflang="es" data-title="Número algebraico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Algebra_nombro" title="Algebra nombro – Esperanto" lang="eo" hreflang="eo" data-title="Algebra nombro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zenbaki_aljebraiko" title="Zenbaki aljebraiko – Basque" lang="eu" hreflang="eu" data-title="Zenbaki aljebraiko" 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/%D8%B9%D8%AF%D8%AF_%D8%AC%D8%A8%D8%B1%DB%8C" title="عدد جبری – Persian" lang="fa" hreflang="fa" data-title="عدد جبری" 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/Nombre_alg%C3%A9brique" title="Nombre algébrique – French" lang="fr" hreflang="fr" data-title="Nombre algébrique" 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/N%C3%BAmero_alx%C3%A9brico" title="Número alxébrico – Galician" lang="gl" hreflang="gl" data-title="Número alxébrico" 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" title="대수적 수 – Korean" lang="ko" hreflang="ko" data-title="대수적 수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%BE%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A9%D5%AB%D5%BE" title="Հանրահաշվական թիվ – Armenian" lang="hy" hreflang="hy" data-title="Հանրահաշվական թիվ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A5%80%E0%A4%AF_%E0%A4%B8%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BE" title="बीजीय संख्या – Hindi" lang="hi" hreflang="hi" data-title="बीजीय संख्या" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Algebarski_broj" title="Algebarski broj – Croatian" lang="hr" hreflang="hr" data-title="Algebarski broj" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Algebrala_nombro" title="Algebrala nombro – Ido" lang="io" hreflang="io" data-title="Algebrala nombro" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Bilangan_aljabar" title="Bilangan aljabar – Indonesian" lang="id" hreflang="id" data-title="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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Algebruleg_tala" title="Algebruleg tala – Icelandic" lang="is" hreflang="is" data-title="Algebruleg tala" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_algebrico" title="Numero algebrico – Italian" lang="it" hreflang="it" data-title="Numero algebrico" 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%9E%D7%A1%D7%A4%D7%A8_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מספר אלגברי – Hebrew" lang="he" hreflang="he" data-title="מספר אלגברי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%BB%D1%8B%D2%9B_%D1%81%D0%B0%D0%BD" title="Алгебралық сан – Kazakh" lang="kk" hreflang="kk" data-title="Алгебралық сан" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%BB%D1%8B%D0%BA_%D1%81%D0%B0%D0%BD" title="Алгебралык сан – Kyrgyz" lang="ky" hreflang="ky" data-title="Алгебралык сан" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Numerus_algebraicus" title="Numerus algebraicus – Latin" lang="la" hreflang="la" data-title="Numerus algebraicus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Algebrisks_skaitlis" title="Algebrisks skaitlis – Latvian" lang="lv" hreflang="lv" data-title="Algebrisks skaitlis" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Algebrinis_skai%C4%8Dius" title="Algebrinis skaičius – Lithuanian" lang="lt" hreflang="lt" data-title="Algebrinis skaičius" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Algebrai_sz%C3%A1m" title="Algebrai szám – Hungarian" lang="hu" hreflang="hu" data-title="Algebrai szám" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Isa_aljebrika" title="Isa aljebrika – Malagasy" lang="mg" hreflang="mg" data-title="Isa aljebrika" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Nombor_algebra" title="Nombor algebra – Malay" lang="ms" hreflang="ms" data-title="Nombor 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%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%9B%E1%80%84%E1%80%BA%E1%80%B8" title="ကိန်းရင်း – Burmese" lang="my" hreflang="my" data-title="ကိန်းရင်း" 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%AFsch_getal" title="Algebraïsch getal – Dutch" lang="nl" hreflang="nl" data-title="Algebraïsch getal" 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%B0" title="代数的数 – Japanese" lang="ja" hreflang="ja" data-title="代数的数" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Algebraiske_tal" title="Algebraiske tal – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Algebraiske tal" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Nombre_algebric" title="Nombre algebric – Occitan" lang="oc" hreflang="oc" data-title="Nombre algebric" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Algebraik_sonlar" title="Algebraik sonlar – Uzbek" lang="uz" hreflang="uz" data-title="Algebraik sonlar" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Liczby_algebraiczne" title="Liczby algebraiczne – Polish" lang="pl" hreflang="pl" data-title="Liczby algebraiczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a 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dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Complex number that is a root of a non-zero polynomial in one variable with rational coefficients</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">Not to be confused with <a href="/wiki/Algebraic_solution" class="mw-redirect" title="Algebraic solution">Algebraic solution</a>.</div> <p class="mw-empty-elt"> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Isosceles_right_triangle_with_legs_length_1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Isosceles_right_triangle_with_legs_length_1.svg/200px-Isosceles_right_triangle_with_legs_length_1.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Isosceles_right_triangle_with_legs_length_1.svg/300px-Isosceles_right_triangle_with_legs_length_1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Isosceles_right_triangle_with_legs_length_1.svg/400px-Isosceles_right_triangle_with_legs_length_1.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>The square root of 2 is an algebraic number equal to the length of the <a href="/wiki/Hypotenuse" title="Hypotenuse">hypotenuse</a> of a <a href="/wiki/Right_triangle" title="Right triangle">right triangle</a> with legs of length 1.</figcaption></figure> <p>An <b>algebraic number</b> is a number that is a <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> of a non-zero <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in one variable with <a href="/wiki/Integer" title="Integer">integer</a> (or, equivalently, <a href="/wiki/Rational_number" title="Rational number">rational</a>) coefficients. For example, the <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</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 (1+{\sqrt {5}})/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+{\sqrt {5}})/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8834bbd69c7f1eddf70573ed4296941c18c25ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.235ex; height:3.009ex;" alt="{\displaystyle (1+{\sqrt {5}})/2}"></span>, is an algebraic number, because it is a root of the polynomial <span class="texhtml"><i>x</i><sup>2</sup> − <i>x</i> − 1</span>. That is, it is a value for x for which the polynomial evaluates to zero. As another example, the <a href="/wiki/Complex_number" title="Complex number">complex number</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 1+i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a65e41a5c0369e908cf26a2452046f19bab946d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle 1+i}"></span> is algebraic because it is a root of <span class="texhtml"><i>x</i><sup>4</sup> + 4</span>. </p><p>All integers and rational numbers are algebraic, as are all <a href="/wiki/Nth_root" title="Nth root">roots of integers</a>. Real and complex numbers that are not algebraic, such as <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> and <span class="texhtml mvar" style="font-style:italic;"><a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a></span>, are called <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental numbers</a>. </p><p>The <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of algebraic (complex) numbers is <a href="/wiki/Countable_set" title="Countable set">countably infinite</a> and has <a href="/wiki/Measure_zero" class="mw-redirect" title="Measure zero">measure zero</a> in the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> as a <a href="/wiki/Subset" title="Subset">subset</a> of the <a href="/wiki/Uncountable_set" title="Uncountable set">uncountable</a> complex numbers. In that sense, <a href="/wiki/Almost_all" title="Almost all">almost all</a> complex numbers are <a href="/wiki/Transcendental_number" title="Transcendental number">transcendental</a>. Similarly, the set of algebraic (real) numbers is countably infinite and has Lebesgue measure zero as a subset of the real numbers, and in that sense almost all real numbers are transcendental. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>All <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> are algebraic. Any rational number, expressed as the quotient of an <a href="/wiki/Integer" title="Integer">integer</a> <span class="texhtml mvar" style="font-style:italic;">a</span> and a (non-zero) <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml mvar" style="font-style:italic;">b</span>, satisfies the above definition, because <span class="texhtml"><i>x</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>a</i></span><span class="sr-only">/</span><span class="den"><i>b</i></span></span>⁠</span></span> is the root of a non-zero polynomial, namely <span class="texhtml"><i>bx</i> − <i>a</i></span>.<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></li> <li><a href="/wiki/Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational numbers</a>, irrational solutions of a quadratic polynomial <span class="texhtml"><i>ax</i><sup>2</sup> + <i>bx</i> + <i>c</i></span> with integer coefficients <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, and <span class="texhtml mvar" style="font-style:italic;">c</span>, are algebraic numbers. If the quadratic polynomial is monic (<span class="texhtml"><i>a</i> = 1</span>), the roots are further qualified as <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a>. <ul><li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a>, complex numbers <span class="texhtml"><i>a</i> + <i>bi</i></span> for which both <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are integers, are also quadratic integers. This is because <span class="texhtml"><i>a</i> + <i>bi</i></span> and <span class="texhtml"><i>a</i> − <i>bi</i></span> are the two roots of the quadratic <span class="texhtml"><i>x</i><sup>2</sup> − 2<i>ax</i> + <i>a</i><sup>2</sup> + <i>b</i><sup>2</sup></span>.</li></ul></li> <li>A <a href="/wiki/Constructible_number" title="Constructible number">constructible number</a> can be constructed from a given unit length using a straightedge and compass. It includes all quadratic irrational roots, all rational numbers, and all numbers that can be formed from these using the <a href="/wiki/Arithmetic_operations" class="mw-redirect" title="Arithmetic operations">basic arithmetic operations</a> and the extraction of square roots. (By designating cardinal directions for +1, −1, +<span class="texhtml mvar" style="font-style:italic;">i</span>, and −<span class="texhtml mvar" style="font-style:italic;">i</span>, complex numbers such as <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 3+i{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3+i{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6995338d180e4c4b0b6946104b883d5f32af1fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.904ex; height:3.009ex;" alt="{\displaystyle 3+i{\sqrt {2}}}"></span> are considered constructible.)</li> <li>Any expression formed from algebraic numbers using any combination of the basic arithmetic operations and extraction of <a href="/wiki/Nth_root" title="Nth root"><span class="texhtml mvar" style="font-style:italic;">n</span>th roots</a> gives another algebraic number.</li> <li>Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of <span class="texhtml mvar" style="font-style:italic;">n</span>th roots (such as the roots of <span class="texhtml"><i>x</i><sup>5</sup> − <i>x</i> + 1</span>). <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">That happens with many</a> but not all polynomials of degree 5 or higher.</li> <li>Values of <a href="/wiki/Trigonometric_functions" title="Trigonometric functions">trigonometric functions</a> of rational multiples of <span class="texhtml mvar" style="font-style:italic;">π</span> (except when undefined): for example, <span class="texhtml">cos <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num"><span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span></span>, <span class="texhtml">cos <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3<span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span></span>, and <span class="texhtml">cos <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5<span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span></span> satisfy <span class="texhtml">8<i>x</i><sup>3</sup> − 4<i>x</i><sup>2</sup> − 4<i>x</i> + 1 = 0</span>. This polynomial is <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over the rationals and so the three cosines are <i>conjugate</i> algebraic numbers. Likewise, <span class="texhtml">tan <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3<span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">16</span></span>⁠</span></span>, <span class="texhtml">tan <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7<span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">16</span></span>⁠</span></span>, <span class="texhtml">tan <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">11<span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">16</span></span>⁠</span></span>, and <span class="texhtml">tan <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">15<span class="texhtml">π</span></span><span class="sr-only">/</span><span class="den">16</span></span>⁠</span></span> satisfy the irreducible polynomial <span class="texhtml"><i>x</i><sup>4</sup> − 4<i>x</i><sup>3</sup> − 6<i>x</i><sup>2</sup> + 4<i>x</i> + 1 = 0</span>, and so are conjugate <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>. This is the equivalent of angles which, when measured in degrees, have rational numbers.<sup id="cite_ref-FOOTNOTEGaribaldi2008_2-0" class="reference"><a href="#cite_note-FOOTNOTEGaribaldi2008-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li>Some but not all irrational numbers are algebraic: <ul><li>The numbers <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 {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></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 {\frac {\sqrt[{3}]{3}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mroot> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </mroot> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\sqrt[{3}]{3}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/844b5ef3eff6de8e73a8df9240ddd6fce05a340a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.934ex; height:5.843ex;" alt="{\displaystyle {\frac {\sqrt[{3}]{3}}{2}}}"></span> are algebraic since they are roots of polynomials <span class="texhtml"><i>x</i><sup>2</sup> − 2</span> and <span class="texhtml">8<i>x</i><sup>3</sup> − 3</span>, respectively.</li> <li>The <a href="/wiki/Golden_ratio" title="Golden ratio">golden ratio</a> <span class="texhtml mvar" style="font-style:italic;">φ</span> is algebraic since it is a root of the polynomial <span class="texhtml"><i>x</i><sup>2</sup> − <i>x</i> − 1</span>.</li> <li>The numbers <a href="/wiki/Pi" title="Pi"><span class="texhtml mvar" style="font-style:italic;">π</span></a> and <a href="/wiki/E_(mathematical_constant)" title="E (mathematical constant)">e</a> are not algebraic numbers (see the <a href="/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a>).<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></li></ul></li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties"><span class="anchor" id="Degree_of_an_algebraic_number"></span> Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Algebraicszoom.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Algebraicszoom.png/220px-Algebraicszoom.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Algebraicszoom.png/330px-Algebraicszoom.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Algebraicszoom.png/440px-Algebraicszoom.png 2x" data-file-width="1920" data-file-height="1080" /></a><figcaption>Algebraic numbers on the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> colored by degree (bright orange/red = 1, green = 2, blue = 3, yellow = 4). The larger points come from polynomials with smaller integer coefficients.</figcaption></figure> <ul><li>If a polynomial with rational coefficients is multiplied through by the <a href="/wiki/Least_common_denominator" class="mw-redirect" title="Least common denominator">least common denominator</a>, the resulting polynomial with integer coefficients has the same roots. This shows that an algebraic number can be equivalently defined as a root of a polynomial with either integer or rational coefficients.</li> <li>Given an algebraic number, there is a unique <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> with rational coefficients of least <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> that has the number as a root. This polynomial is called its <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a>. If its minimal polynomial has degree <span class="texhtml mvar" style="font-style:italic;">n</span>, then the algebraic number is said to be of <b>degree <span class="texhtml mvar" style="font-style:italic;">n</span></b>. For example, all <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> have degree 1, and an algebraic number of degree 2 is a <a href="/wiki/Quadratic_irrational" class="mw-redirect" title="Quadratic irrational">quadratic irrational</a>.</li> <li>The algebraic numbers are <a href="/wiki/Dense_set" title="Dense set">dense</a> <a href="/wiki/Densely_ordered" class="mw-redirect" title="Densely ordered">in the reals</a>. This follows from the fact they contain the rational numbers, which are dense in the reals themselves.</li> <li>The set of algebraic numbers is countable (enumerable),<sup id="cite_ref-FOOTNOTEHardyWright19721602008:205_4-0" class="reference"><a href="#cite_note-FOOTNOTEHardyWright19721602008:205-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTENiven1956Theorem_7.5._5-0" class="reference"><a href="#cite_note-FOOTNOTENiven1956Theorem_7.5.-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and therefore its <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a> as a subset of the complex numbers is 0 (essentially, the algebraic numbers take up no space in the complex numbers). That is to say, <a href="/wiki/Almost_everywhere" title="Almost everywhere">"almost all"</a> real and complex numbers are transcendental.</li> <li>All algebraic numbers are <a href="/wiki/Computable_number" title="Computable number">computable</a> and therefore <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable</a> and <a href="/wiki/Arithmetical_numbers" class="mw-redirect" title="Arithmetical numbers">arithmetical</a>.</li> <li>For real numbers <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span>, the complex number <span class="texhtml"><i>a</i> + <i>bi</i></span> is algebraic if and only if both <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are algebraic.<sup id="cite_ref-FOOTNOTENiven1956Corollary_7.3._6-0" class="reference"><a href="#cite_note-FOOTNOTENiven1956Corollary_7.3.-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity">Degree of simple extensions of the rationals as a criterion to algebraicity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=3" title="Edit section: Degree of simple extensions of the rationals as a criterion to algebraicity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <span class="texhtml">α</span>, the <a href="/wiki/Simple_extension" title="Simple extension">simple extension</a> of the rationals by <span class="texhtml">α</span>, denoted 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 \mathbb {Q} (\alpha )\equiv \{\sum _{i=-{n_{1}}}^{n_{2}}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q} ,n_{1},n_{2}\in \mathbb {N} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>≡</mo> <mo fence="false" stretchy="false">{</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </munderover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )\equiv \{\sum _{i=-{n_{1}}}^{n_{2}}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q} ,n_{1},n_{2}\in \mathbb {N} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b5da5f4ff78c9d9ba4f7c68f117650407316188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:38.599ex; height:7.343ex;" alt="{\displaystyle \mathbb {Q} (\alpha )\equiv \{\sum _{i=-{n_{1}}}^{n_{2}}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q} ,n_{1},n_{2}\in \mathbb {N} \}}"></span>, is of finite <a href="/wiki/Degree_of_a_field_extension" title="Degree of a field extension">degree</a> if and only if <span class="texhtml">α</span> is an algebraic number. </p><p>The condition of finite degree means that there is a finite set <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_{i}|1\leq i\leq k\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>k</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{i}|1\leq i\leq k\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b992d13b0c5842236e425739b3a8f05f73ce2c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.374ex; height:2.843ex;" alt="{\displaystyle \{a_{i}|1\leq i\leq k\}}"></span> 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 \mathbb {Q} (\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bedf30cf0700ad5765c98e3eeccf3ec85a655d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.105ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\alpha )}"></span> such that <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} (\alpha )=\sum _{i=1}^{k}a_{i}\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <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>k</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )=\sum _{i=1}^{k}a_{i}\mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd15ce285f8df7519e1db023ee0ed08e82fabb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.783ex; height:7.343ex;" alt="{\displaystyle \mathbb {Q} (\alpha )=\sum _{i=1}^{k}a_{i}\mathbb {Q} }"></span>; that is, every member 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 \mathbb {Q} (\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bedf30cf0700ad5765c98e3eeccf3ec85a655d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.105ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\alpha )}"></span> can be written as <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 _{i=1}^{k}a_{i}q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{k}a_{i}q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89bd018c5a882b9855e82f346d63b2a5722eaf19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:7.608ex; height:7.343ex;" alt="{\displaystyle \sum _{i=1}^{k}a_{i}q_{i}}"></span> for some rational numbers <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_{i}|1\leq i\leq k\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>k</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{q_{i}|1\leq i\leq k\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c8d658e3cae47ac1b72019d36307dc43f0ac06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.182ex; height:2.843ex;" alt="{\displaystyle \{q_{i}|1\leq i\leq k\}}"></span> (note that the set <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_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/434434a3a4c297856e0eff9f57d2d25053f830b7" 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 \{a_{i}\}}"></span> is fixed). </p><p>Indeed, since 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 a_{i}-s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}-s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fe1151c07a91e5353e9916fe4baa4a18f6d8f1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.96ex; height:2.343ex;" alt="{\displaystyle a_{i}-s}"></span> are themselves members 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 \mathbb {Q} (\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bedf30cf0700ad5765c98e3eeccf3ec85a655d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.105ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\alpha )}"></span>, each can be expressed as sums of products of rational numbers and powers of <span class="texhtml">α</span>, and therefore this condition is equivalent to the requirement that for some finite <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, <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} (\alpha )=\{\sum _{i=-n}^{n}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )=\{\sum _{i=-n}^{n}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/622b97f6349a04e77c8512e26ff5e17473dab4c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.283ex; height:7.009ex;" alt="{\displaystyle \mathbb {Q} (\alpha )=\{\sum _{i=-n}^{n}\alpha ^{i}q_{i}|q_{i}\in \mathbb {Q} \}}"></span>. </p><p>The latter condition is equivalent 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 \alpha ^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{n+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e1a135c0da5d0d28280d11ef1e7a7096170f74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.807ex; height:2.676ex;" alt="{\displaystyle \alpha ^{n+1}}"></span>, itself a member 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 \mathbb {Q} (\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bedf30cf0700ad5765c98e3eeccf3ec85a655d55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.105ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\alpha )}"></span>, being expressible as <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 _{i=-n}^{n}\alpha ^{i}q_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=-n}^{n}\alpha ^{i}q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8d38226ce02ef6fc5e1a4817f7ef3bc6b9091c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.622ex; height:7.009ex;" alt="{\displaystyle \sum _{i=-n}^{n}\alpha ^{i}q_{i}}"></span> for some rationals <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_{i}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{q_{i}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21aa31eadcb8a3e97a21b1226a18fa08cd9fcb37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.162ex; height:2.843ex;" alt="{\displaystyle \{q_{i}\}}"></span>, so <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 \alpha ^{2n+1}=\sum _{i=0}^{2n}\alpha ^{i}q_{i-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ^{2n+1}=\sum _{i=0}^{2n}\alpha ^{i}q_{i-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cfb258d9f9d731739551e42a2f498c660f5154b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.858ex; height:7.343ex;" alt="{\displaystyle \alpha ^{2n+1}=\sum _{i=0}^{2n}\alpha ^{i}q_{i-n}}"></span> or, equivalently, <span class="texhtml">α</span> is a root 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 x^{2n+1}-\sum _{i=0}^{2n}x^{i}q_{i-n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </munderover> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2n+1}-\sum _{i=0}^{2n}x^{i}q_{i-n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a94392298831eb9e505ea1e25d6ff89dcb5e89e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:18.284ex; height:7.343ex;" alt="{\displaystyle x^{2n+1}-\sum _{i=0}^{2n}x^{i}q_{i-n}}"></span>; that is, an algebraic number with a minimal polynomial of degree not larger than <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 2n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2n+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca410f731fe4c7c444330343afb1d1850eadaea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.56ex; height:2.343ex;" alt="{\displaystyle 2n+1}"></span>. </p><p>It can similarly be proven that for any finite set of algebraic numbers <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 \alpha _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d69430689b8aac2832684109cde587c4ae828d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{1}}"></span>, <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 \alpha _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef5f08a56f51deb324e5eed2fb9b2e3c279889d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{2}}"></span>... <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 \alpha _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87bc96d36d1f4d3c81e942c864b212b2ab50b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.706ex; height:2.009ex;" alt="{\displaystyle \alpha _{n}}"></span>, the field 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 \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243815c52e38fd8d543d3a5c3dabc3d41cdc0f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.577ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})}"></span> has a finite degree. </p> <div class="mw-heading mw-heading2"><h2 id="Field">Field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=4" title="Edit section: Field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Algebraic_number_in_the_complex_plane.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Algebraic_number_in_the_complex_plane.png/220px-Algebraic_number_in_the_complex_plane.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Algebraic_number_in_the_complex_plane.png/330px-Algebraic_number_in_the_complex_plane.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Algebraic_number_in_the_complex_plane.png/440px-Algebraic_number_in_the_complex_plane.png 2x" data-file-width="779" data-file-height="516" /></a><figcaption>Algebraic numbers colored by degree (blue = 4, cyan = 3, red = 2, green = 1). The unit circle is black.<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="What does this figure tell us about algebraic numbers? Can we get some insight out of it, or it this just mathematical art? (July 2024)">further explanation needed</span></a></i>]</sup></figcaption></figure> <p>The sum, difference, product, and quotient (if the denominator is nonzero) of two algebraic numbers is again algebraic: </p><p>For any two algebraic numbers <span class="texhtml">α</span>, <span class="texhtml">β</span>, this follows directly from the fact that the <a href="/wiki/Simple_extension" title="Simple extension">simple extension</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} (\gamma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\gamma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db05203a7c9b1e40c0770162b0b52c8d8cbf7ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\gamma )}"></span>, for <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> being either <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 \alpha +\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha +\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99b80a3fdecb9cf75091789bb4335a1bb3561b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha +\beta }"></span>, <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 \alpha -\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>−</mo> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha -\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9836cb70870377fca13c9f0808126dd7909270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.66ex; height:2.509ex;" alt="{\displaystyle \alpha -\beta }"></span>, <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 \alpha \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2efb0e5523f52275f3193b0dfd9a92ad5b76830c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.509ex;" alt="{\displaystyle \alpha \beta }"></span> or (for <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 \beta \neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6347ae2e04e6c0cb5c10d5ec3266a67f9e54da6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.593ex; height:2.676ex;" alt="{\displaystyle \beta \neq 0}"></span>) <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 \alpha /\beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha /\beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/216fe59007cdea29a4e023a1af39f740cb5a2b65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.982ex; height:2.843ex;" alt="{\displaystyle \alpha /\beta }"></span>, is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of the finite-<a href="/wiki/Degree_of_a_field_extension" title="Degree of a field extension">degree</a> field 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 \mathbb {Q} (\alpha ,\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\alpha ,\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6615478c0d5353d1f187642d3d21e012aa0f6c76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.471ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\alpha ,\beta )}"></span>, and therefore has a finite degree itself, from which it follows (as shown <a href="#Degree_of_simple_extensions_of_the_rationals_as_a_criterion_to_algebraicity">above</a>) that <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is algebraic. </p><p>An alternative way of showing this is constructively, by using the <a href="/wiki/Resultant" title="Resultant">resultant</a>. </p><p>Algebraic numbers thus form a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a><sup id="cite_ref-FOOTNOTENiven195692_7-0" class="reference"><a href="#cite_note-FOOTNOTENiven195692-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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 {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a8814b1ca454c488e409813988dd5dd906148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.923ex; height:3.343ex;" alt="{\displaystyle {\overline {\mathbb {Q} }}}"></span> (sometimes denoted 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 \mathbb {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>, but that usually denotes the <a href="/wiki/Adele_ring" title="Adele ring">adele ring</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_closure">Algebraic closure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=5" title="Edit section: Algebraic closure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every root of a polynomial equation whose coefficients are <i>algebraic numbers</i> is again algebraic. That can be rephrased by saying that the field of algebraic numbers is <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>. In fact, it is the smallest algebraically closed field containing the rationals and so it is called the <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of the rationals. </p><p>That the field of algebraic numbers is algebraically closed can be proven as follows: Let <span class="texhtml">β</span> be a root of a polynomial <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 \alpha _{0}+\alpha _{1}x+\alpha _{2}x^{2}...+\alpha _{n}x^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{0}+\alpha _{1}x+\alpha _{2}x^{2}...+\alpha _{n}x^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a923751a914879db26bbc8f674ddbb969f122542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.184ex; height:3.009ex;" alt="{\displaystyle \alpha _{0}+\alpha _{1}x+\alpha _{2}x^{2}...+\alpha _{n}x^{n}}"></span> with coefficients that are algebraic numbers <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 \alpha _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a214eff2fcc322f780dd8837e7472b0edb994a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{0}}"></span>, <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 \alpha _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d69430689b8aac2832684109cde587c4ae828d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{1}}"></span>, <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 \alpha _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef5f08a56f51deb324e5eed2fb9b2e3c279889d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{2}}"></span>... <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 \alpha _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87bc96d36d1f4d3c81e942c864b212b2ab50b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.706ex; height:2.009ex;" alt="{\displaystyle \alpha _{n}}"></span>. The field 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 \mathbb {Q} ^{\prime }\equiv \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{\prime }\equiv \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90f63d8bed52beb31cbb5393037edfa3fcd5da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.168ex; height:3.009ex;" alt="{\displaystyle \mathbb {Q} ^{\prime }\equiv \mathbb {Q} (\alpha _{1},\alpha _{2},...\alpha _{n})}"></span> then has a finite degree with respect 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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>. The simple 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 \mathbb {Q} ^{\prime }(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{\prime }(\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4b528b22bd3c6dac904127f8dd8530be27f236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.634ex; height:3.009ex;" alt="{\displaystyle \mathbb {Q} ^{\prime }(\beta )}"></span> then has a finite degree with respect 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 \mathbb {Q} ^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b3f9d3ccb1a550c79cad8f06b98edf0d90b8f96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.493ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} ^{\prime }}"></span> (since all powers of <span class="texhtml">β</span> can be expressed by powers of up 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 \beta ^{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta ^{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e42241e5e251940bab6374c95c6ff9ea8682bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.656ex; height:3.009ex;" alt="{\displaystyle \beta ^{n-1}}"></span>). Therefore, <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} ^{\prime }(\beta )=\mathbb {Q} (\beta ,\alpha _{1},\alpha _{2},...\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{\prime }(\beta )=\mathbb {Q} (\beta ,\alpha _{1},\alpha _{2},...\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199787526e6f6500145eb1508fcfee0d4577455b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.675ex; height:3.009ex;" alt="{\displaystyle \mathbb {Q} ^{\prime }(\beta )=\mathbb {Q} (\beta ,\alpha _{1},\alpha _{2},...\alpha _{n})}"></span> also has a finite degree with respect 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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>. Since <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} (\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} (\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7821dad9921ba34f868fc950bf26a0f4a27adfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.949ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} (\beta )}"></span> is a linear subspace 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 \mathbb {Q} ^{\prime }(\beta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>β</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{\prime }(\beta )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc4b528b22bd3c6dac904127f8dd8530be27f236" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.634ex; height:3.009ex;" alt="{\displaystyle \mathbb {Q} ^{\prime }(\beta )}"></span>, it must also have a finite degree with respect 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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>, so <span class="texhtml">β</span> must be an algebraic number. </p> <div class="mw-heading mw-heading2"><h2 id="Related_fields">Related fields</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=6" title="Edit section: Related fields"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Numbers_defined_by_radicals">Numbers defined by radicals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=7" title="Edit section: Numbers defined by radicals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any number that can be obtained from the integers using a <a href="/wiki/Finite_set" title="Finite set">finite</a> number of <a href="/wiki/Addition" title="Addition">additions</a>, <a href="/wiki/Subtraction" title="Subtraction">subtractions</a>, <a href="/wiki/Multiplication" title="Multiplication">multiplications</a>, <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">divisions</a>, and taking (possibly complex) <span class="texhtml mvar" style="font-style:italic;">n</span>th roots where <span class="texhtml mvar" style="font-style:italic;">n</span> is a positive integer are algebraic. The converse, however, is not true: there are algebraic numbers that cannot be obtained in this manner. These numbers are roots of polynomials of degree 5 or higher, a result of <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> (see <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">Quintic equations</a> and the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a>). For example, the equation: </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^{5}-x-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{5}-x-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f0300136d28f9da40ab77b69ead236e1e8277eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.818ex; height:2.843ex;" alt="{\displaystyle x^{5}-x-1=0}"></span></dd></dl> <p>has a unique real root that cannot be expressed in terms of only radicals and arithmetic operations. </p> <div class="mw-heading mw-heading3"><h3 id="Closed-form_number">Closed-form number</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=8" title="Edit section: Closed-form number"><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/Closed-form_number" class="mw-redirect" title="Closed-form number">Closed-form number</a></div> <p>Algebraic numbers are all numbers that can be defined explicitly or implicitly in terms of polynomials, starting from the rational numbers. One may generalize this to "<a href="/wiki/Closed-form_number" class="mw-redirect" title="Closed-form number">closed-form numbers</a>", which may be defined in various ways. Most broadly, all numbers that can be defined explicitly or implicitly in terms of polynomials, exponentials, and logarithms are called "<a href="/wiki/Elementary_number" title="Elementary number">elementary numbers</a>", and these include the algebraic numbers, plus some transcendental numbers. Most narrowly, one may consider numbers <i>explicitly</i> defined in terms of polynomials, exponentials, and logarithms – this does not include all algebraic numbers, but does include some simple transcendental numbers such as <span class="texhtml mvar" style="font-style:italic;">e</span> or <a href="/wiki/Natural_logarithm_of_2" title="Natural logarithm of 2">ln 2</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Algebraic_integers">Algebraic integers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=9" title="Edit section: Algebraic integers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Leadingcoeff.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Leadingcoeff.png/220px-Leadingcoeff.png" decoding="async" width="220" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Leadingcoeff.png/330px-Leadingcoeff.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Leadingcoeff.png/440px-Leadingcoeff.png 2x" data-file-width="1920" data-file-height="1080" /></a><figcaption>Visualisation of the (countable) field of algebraic numbers in the complex plane. Colours indicate the leading integer coefficient of the polynomial the number is a root of (red = 1 i.e. the algebraic integers, green = 2, blue = 3, yellow = 4...). Points becomes smaller as the other coefficients and number of terms in the polynomial become larger. View shows integers 0,1 and 2 at bottom right, +i near top.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_integer" title="Algebraic integer">Algebraic integer</a></div> <p>An <i>algebraic integer</i> is an algebraic number that is a root of a polynomial with integer coefficients with leading coefficient 1 (a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a>). Examples of algebraic integers are <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+13{\sqrt {2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>+</mo> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5+13{\sqrt {2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f1e9b07e86cbb4134f23efab66df7124ca5195d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.073ex; height:3.009ex;" alt="{\displaystyle 5+13{\sqrt {2}},}"></span> <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 2-6i,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>−</mo> <mn>6</mn> <mi>i</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2-6i,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ddcc9007e1a41ddeba514402085feaad1b3633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.615ex; height:2.509ex;" alt="{\displaystyle 2-6i,}"></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="{\textstyle {\frac {1}{2}}(1+i{\sqrt {3}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}(1+i{\sqrt {3}}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8962be8e99592140410cdbba97852b5db9490623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.018ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}(1+i{\sqrt {3}}).}"></span> Therefore, the algebraic integers constitute a proper <a href="/wiki/Superset" class="mw-redirect" title="Superset">superset</a> of the <a href="/wiki/Integer" title="Integer">integers</a>, as the latter are the roots of monic polynomials <span class="texhtml"><i>x</i> − <i>k</i></span> for all <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 {Z} }"> <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">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a12237af5f2ec5fc7c5023f439266bae1380f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.602ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {Z} }"></span>. In this sense, algebraic integers are to algebraic numbers what <a href="/wiki/Integer" title="Integer">integers</a> are to <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>. </p><p>The sum, difference and product of algebraic integers are again algebraic integers, which means that the algebraic integers form a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>. The name <i>algebraic integer</i> comes from the fact that the only rational numbers that are algebraic integers are the integers, and because the algebraic integers in any <a href="/wiki/Algebraic_number_field" title="Algebraic number field">number field</a> are in many ways analogous to the integers. If <span class="texhtml"><i>K</i></span> is a number field, its <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> is the subring of algebraic integers in <span class="texhtml"><i>K</i></span>, and is frequently denoted as <span class="texhtml"><i>O<sub>K</sub></i></span>. These are the prototypical examples of <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domains</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Special_classes">Special classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=10" title="Edit section: Special classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_solution" class="mw-redirect" title="Algebraic solution">Algebraic solution</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li> <li><a href="/wiki/Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational number</a></li> <li><a href="/wiki/Fundamental_unit_(number_theory)" title="Fundamental unit (number theory)">Fundamental unit</a></li> <li><a href="/wiki/Root_of_unity" title="Root of unity">Root of unity</a></li> <li><a href="/wiki/Gaussian_period" title="Gaussian period">Gaussian period</a></li> <li><a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a></li> <li><a href="/wiki/Salem_number" title="Salem number">Salem 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&action=edit&section=11" 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"><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">Some of the following examples come from <a href="#CITEREFHardyWright1972">Hardy & Wright (1972</a>, pp. 159–160, 178–179)</span> </li> <li id="cite_note-FOOTNOTEGaribaldi2008-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGaribaldi2008_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGaribaldi2008">Garibaldi 2008</a>.</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">Also, <a href="/wiki/Liouville_number" title="Liouville number">Liouville's theorem</a> can be used to "produce as many examples of transcendental numbers as we please," cf. <a href="#CITEREFHardyWright1972">Hardy & Wright (1972</a>, p. 161ff)</span> </li> <li id="cite_note-FOOTNOTEHardyWright19721602008:205-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHardyWright19721602008:205_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHardyWright1972">Hardy & Wright 1972</a>, p. 160, 2008:205.</span> </li> <li id="cite_note-FOOTNOTENiven1956Theorem_7.5.-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENiven1956Theorem_7.5._5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNiven1956">Niven 1956</a>, Theorem 7.5..</span> </li> <li id="cite_note-FOOTNOTENiven1956Corollary_7.3.-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENiven1956Corollary_7.3._6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNiven1956">Niven 1956</a>, Corollary 7.3..</span> </li> <li id="cite_note-FOOTNOTENiven195692-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTENiven195692_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFNiven1956">Niven 1956</a>, p. 92.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_number&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFArtin1991" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (1991), <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/algebra0000arti_x4a1/"><i>Algebra</i></a></span>, Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-004763-5" title="Special:BookSources/0-13-004763-5"><bdi>0-13-004763-5</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=1129886">1129886</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pub=Prentice+Hall&rft.date=1991&rft.isbn=0-13-004763-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1129886%23id-name%3DMR&rft.aulast=Artin&rft.aufirst=Michael&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebra0000arti_x4a1%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGaribaldi2008" class="citation cs2">Garibaldi, Skip (June 2008), "Somewhat more than governors need to know about trigonometry", <i>Mathematics Magazine</i>, <b>81</b> (3): 191–200, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0025570x.2008.11953548">10.1080/0025570x.2008.11953548</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/27643106">27643106</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Somewhat+more+than+governors+need+to+know+about+trigonometry&rft.volume=81&rft.issue=3&rft.pages=191-200&rft.date=2008-06&rft_id=info%3Adoi%2F10.1080%2F0025570x.2008.11953548&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F27643106%23id-name%3DJSTOR&rft.aulast=Garibaldi&rft.aufirst=Skip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHardyWright1972" class="citation cs2"><a href="/wiki/G._H._Hardy" title="G. H. Hardy">Hardy, Godfrey Harold</a>; <a href="/wiki/E._M._Wright" title="E. M. Wright">Wright, Edward M.</a> (1972), <i>An introduction to the theory of numbers</i> (5th ed.), Oxford: Clarendon, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853171-0" title="Special:BookSources/0-19-853171-0"><bdi>0-19-853171-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+theory+of+numbers&rft.place=Oxford&rft.edition=5th&rft.pub=Clarendon&rft.date=1972&rft.isbn=0-19-853171-0&rft.aulast=Hardy&rft.aufirst=Godfrey+Harold&rft.au=Wright%2C+Edward+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIrelandRosen1990" class="citation cs2">Ireland, Kenneth; Rosen, Michael (1990) [1st ed. 1982], <i>A Classical Introduction to Modern Number Theory</i> (2nd ed.), Berlin: 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/0-387-97329-X" title="Special:BookSources/0-387-97329-X"><bdi>0-387-97329-X</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=1070716">1070716</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Classical+Introduction+to+Modern+Number+Theory&rft.place=Berlin&rft.edition=2nd&rft.pub=Springer&rft.date=1990&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1070716%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-2103-4&rft.isbn=0-387-97329-X&rft.aulast=Ireland&rft.aufirst=Kenneth&rft.au=Rosen%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2002" class="citation cs2">Lang, Serge (2002) [1st ed. 1965], <a rel="nofollow" class="external text" href="https://archive.org/details/algebra-serge-lang/"><i>Algebra</i></a> (3rd ed.), New York: Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95385-4" title="Special:BookSources/978-0-387-95385-4"><bdi>978-0-387-95385-4</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=1878556">1878556</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York&rft.edition=3rd&rft.pub=Springer&rft.date=2002&rft.isbn=978-0-387-95385-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1878556%23id-name%3DMR&rft.aulast=Lang&rft.aufirst=Serge&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Falgebra-serge-lang%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNiven1956" class="citation cs2"><a href="/wiki/Ivan_M._Niven" title="Ivan M. Niven">Niven, Ivan M.</a> (1956), <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/irrationalnumber00nive/"><i>Irrational Numbers</i></a></span>, <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Irrational+Numbers&rft.pub=Mathematical+Association+of+America&rft.date=1956&rft.aulast=Niven&rft.aufirst=Ivan+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Firrationalnumber00nive%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOre1948" class="citation cs2"><a href="/wiki/%C3%98ystein_Ore" title="Øystein Ore">Ore, Øystein</a> (1948), <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/numbertheoryitsh00ore/"><i>Number Theory and Its History</i></a></span>, New York: McGraw-Hill</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Number+Theory+and+Its+History&rft.place=New+York&rft.pub=McGraw-Hill&rft.date=1948&rft.aulast=Ore&rft.aufirst=%C3%98ystein&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnumbertheoryitsh00ore%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+number" class="Z3988"></span></li></ul> <div class="navbox-styles"><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 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number</a></li> <li><a href="/wiki/Look-and-say_sequence" title="Look-and-say sequence">Conway's constant</a></li> <li><a href="/wiki/Cyclotomic_field" title="Cyclotomic field">Cyclotomic field</a></li> <li><a href="/wiki/Doubling_the_cube" title="Doubling the cube">Doubling the cube</a></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integer</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li> <li><a href="/wiki/Golden_ratio" title="Golden ratio">Golden ratio (<span class="texhtml mvar" style="font-style:italic;">φ</span>)</a></li> <li><a href="/wiki/Perron_number" title="Perron number">Perron number</a></li> <li><a href="/wiki/Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a></li> <li><a href="/wiki/Plastic_ratio" title="Plastic ratio">Plastic ratio (<span class="texhtml mvar" style="font-style:italic;">ρ</span>)</a></li> <li><a href="/wiki/Quadratic_irrational_number" title="Quadratic irrational number">Quadratic irrational number</a></li> <li><a href="/wiki/Rational_number" title="Rational number">Rational number</a></li> <li><a href="/wiki/Root_of_unity" title="Root of unity">Root of unity</a></li> <li><a href="/wiki/Salem_number" title="Salem number">Salem number</a></li> <li><a href="/wiki/Silver_ratio" title="Silver ratio">Silver ratio (<span class="texhtml mvar" style="font-style:italic;">δ</span><sub><span class="texhtml mvar" style="font-style:italic;">S</span></sub>)</a></li> <li><a href="/wiki/Square_root_of_2" title="Square root of 2">Square root of 2</a></li> <li><a href="/wiki/Square_root_of_3" title="Square root of 3">Square root of 3</a></li> <li><a href="/wiki/Square_root_of_5" title="Square root of 5">Square root of 5</a></li> <li><a href="/wiki/Square_root_of_6" title="Square root of 6">Square root of 6</a></li> <li><a href="/wiki/Square_root_of_7" title="Square root of 7">Square root of 7</a></li> <li><a href="/wiki/Supergolden_ratio" title="Supergolden ratio">Supergolden ratio (<span class="texhtml mvar" style="font-style:italic;">ψ</span>)</a></li> <li><a href="/wiki/Supersilver_ratio" title="Supersilver ratio">Supersilver ratio (<span class="texhtml mvar" style="font-style:italic;">ς</span>)</a></li> <li><a href="/wiki/Twelfth_root_of_2" class="mw-redirect" title="Twelfth root of 2">Twelfth root of 2</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Number_systems" 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"><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_systems" title="Template:Number systems"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Number_systems" title="Template talk:Number systems"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Number_systems" title="Special:EditPage/Template:Number systems"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Number_systems" style="font-size:114%;margin:0 4em"><a href="/wiki/Number" title="Number">Number</a> systems</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sets of <a href="/wiki/Definable_number" class="mw-redirect" title="Definable number">definable numbers</a></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/Natural_number" title="Natural number">Natural 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 {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>)</li> <li><a href="/wiki/Integer" title="Integer">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} }"> <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>)</li> <li><a href="/wiki/Rational_number" title="Rational number">Rational 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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span>)</li> <li><a href="/wiki/Constructible_number" title="Constructible number">Constructible numbers</a></li> <li><a class="mw-selflink selflink">Algebraic 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 {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fb423c16a5f403edbaf66438b75e7a36e725af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {A} }"></span>)</li> <li><a href="/wiki/Closed-form_expression#Closed-form_number" title="Closed-form expression">Closed-form numbers</a></li> <li><a href="/wiki/Period_(algebraic_geometry)" title="Period (algebraic geometry)">Periods</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 {\mathcal {P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d6ec962de5797ba4f161c40e66dca74ae95cc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.704ex; height:2.176ex;" alt="{\displaystyle {\mathcal {P}}}"></span>)</li> <li><a href="/wiki/Computable_number" title="Computable number">Computable numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_arithmetic" title="Definable real number">Arithmetical numbers</a></li> <li><a href="/wiki/Definable_real_number#Definability_in_models_of_ZFC" title="Definable real number">Set-theoretically definable numbers</a></li> <li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> <ul><li><a href="/wiki/Gaussian_rational" title="Gaussian rational">Gaussian rationals</a></li></ul></li> <li><a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebras</a></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/Division_algebra" title="Division algebra">Division algebras</a>: <a href="/wiki/Real_number" title="Real number">Real 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 {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>)</li> <li><a href="/wiki/Complex_number" title="Complex number">Complex 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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>)</li> <li><a href="/wiki/Quaternion" title="Quaternion">Quaternions</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 {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e050965453c42bcc6bd544546703c836bdafeac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} }"></span>)</li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</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 {O} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {O} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ed2664a4fe515e6fbed25a7193ce663b82920c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \mathbb {O} }"></span>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Split<br />types</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>Over <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 {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>:</li> <li><a href="/wiki/Split-complex_number" title="Split-complex number">Split-complex numbers</a></li> <li><a href="/wiki/Split-quaternion" title="Split-quaternion">Split-quaternions</a></li> <li><a href="/wiki/Split-octonion" title="Split-octonion">Split-octonions</a><br /> Over <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>:</li> <li><a href="/wiki/Bicomplex_number" title="Bicomplex number">Bicomplex numbers</a></li> <li><a href="/wiki/Biquaternion" title="Biquaternion">Biquaternions</a></li> <li><a href="/wiki/Bioctonion" title="Bioctonion">Bioctonions</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other <a href="/wiki/Hypercomplex_number" title="Hypercomplex number">hypercomplex</a></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/Dual_number" title="Dual number">Dual numbers</a></li> <li><a href="/wiki/Dual_quaternion" title="Dual quaternion">Dual quaternions</a></li> <li><a href="/wiki/Dual-complex_number" class="mw-redirect" title="Dual-complex number">Dual-complex numbers</a></li> <li><a href="/wiki/Hyperbolic_quaternion" title="Hyperbolic quaternion">Hyperbolic quaternions</a></li> <li><a href="/wiki/Sedenion" title="Sedenion">Sedenions</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 {S} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f9d5874c5d7f68eba1cec9da9ccbe53903303bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.293ex; height:2.176ex;" alt="{\displaystyle \mathbb {S} }"></span>)</li> <li><a href="/wiki/Trigintaduonion" title="Trigintaduonion">Trigintaduonions</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 {T} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">T</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {T} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c039979935c00b3b216cbb065999207872677f6" 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 {T} }"></span>)</li> <li><a href="/wiki/Split-biquaternion" title="Split-biquaternion">Split-biquaternions</a></li> <li><a href="/wiki/Multicomplex_number" title="Multicomplex number">Multicomplex numbers</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a>/<a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <ul><li><a href="/wiki/Algebra_of_physical_space" title="Algebra of physical space">Algebra of physical space</a></li> <li><a href="/wiki/Spacetime_algebra" title="Spacetime algebra">Spacetime algebra</a></li> <li><a href="/wiki/Plane-based_geometric_algebra" title="Plane-based geometric algebra">Plane-based geometric algebra</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Infinity" title="Infinity">Infinities</a> and <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimals</a></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/Cardinal_number" title="Cardinal number">Cardinal numbers</a></li> <li><a href="/wiki/Extended_natural_numbers" title="Extended natural numbers">Extended natural numbers</a></li> <li><a href="/wiki/Extended_real_number_line" title="Extended real number line">Extended real numbers</a> <ul><li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projective</a></li></ul></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Extended complex numbers</a></li> <li><a href="/wiki/Hyperreal_number" title="Hyperreal number">Hyperreal numbers</a></li> <li><a href="/wiki/Levi-Civita_field" title="Levi-Civita field">Levi-Civita field</a></li> <li><a href="/wiki/Ordinal_number" title="Ordinal number">Ordinal numbers</a></li> <li><a href="/wiki/Supernatural_number" title="Supernatural number">Supernatural numbers</a></li> <li><a href="/wiki/Surreal_number" title="Surreal number">Surreal numbers</a></li> <li><a href="/wiki/Superreal_number" title="Superreal number">Superreal numbers</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other types</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/Irrational_number" title="Irrational number">Irrational numbers</a></li> <li><a href="/wiki/Fuzzy_number" title="Fuzzy number">Fuzzy 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"><span class="nowrap"><i>p</i>-adic</span> numbers</a> (<a href="/wiki/Solenoid_(mathematics)#p-adic_solenoids" title="Solenoid (mathematics)"><span class="nowrap"><i>p</i>-adic</span> solenoids</a>)</li> <li><a href="/wiki/Profinite_integer" title="Profinite integer">Profinite integers</a></li> <li><a href="/wiki/Normal_number" title="Normal number">Normal numbers</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2"><div> <ul><li><a href="/wiki/Number#Main_classification" title="Number">Classification</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, 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