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href="/wiki/Especial:A_mi%C3%B1a_conversa" title="Conversa acerca de edicións feitas desde este enderezo IP [n]" accesskey="n"><span>Conversa</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Sitio"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contidos" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover á barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">agochar</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Inicio</div> </a> </li> <li id="toc-Definición" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definición"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definición</span> </div> </a> <ul id="toc-Definición-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemplos_de_corpos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemplos_de_corpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exemplos de corpos</span> </div> </a> <button aria-controls="toc-Exemplos_de_corpos-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>Mostrar ou agochar a subsección "Exemplos de corpos"</span> </button> <ul id="toc-Exemplos_de_corpos-sublist" class="vector-toc-list"> <li id="toc-Racionais_e_alxébricos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Racionais_e_alxébricos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Racionais e alxébricos</span> </div> </a> <ul id="toc-Racionais_e_alxébricos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Números_reais,_complexos_e_p-ádicos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Números_reais,_complexos_e_p-ádicos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Números reais, complexos e <i>p</i>-ádicos</span> </div> </a> <ul id="toc-Números_reais,_complexos_e_p-ádicos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corpos_finitos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Corpos_finitos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Corpos finitos</span> </div> </a> <ul id="toc-Corpos_finitos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corpos_de_funcións" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Corpos_de_funcións"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Corpos de funcións</span> </div> </a> <ul id="toc-Corpos_de_funcións-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ultrafiltros" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ultrafiltros"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Ultrafiltros</span> </div> </a> <ul id="toc-Ultrafiltros-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subcorpos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subcorpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>Subcorpos</span> </div> </a> <ul id="toc-Subcorpos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algúns_teoremas_iniciais" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algúns_teoremas_iniciais"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algúns teoremas iniciais</span> </div> </a> <ul id="toc-Algúns_teoremas_iniciais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construcións_de_corpos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Construcións_de_corpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Construcións de corpos</span> </div> </a> <button aria-controls="toc-Construcións_de_corpos-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>Mostrar ou agochar a subsección "Construcións de corpos"</span> </button> <ul id="toc-Construcións_de_corpos-sublist" class="vector-toc-list"> <li id="toc-Subcorpos_e_ideais" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subcorpos_e_ideais"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Subcorpos e ideais</span> </div> </a> <ul id="toc-Subcorpos_e_ideais-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corpo_de_fraccións" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Corpo_de_fraccións"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Corpo de fraccións</span> </div> </a> <ul id="toc-Corpo_de_fraccións-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensión_de_corpos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extensión_de_corpos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Extensión de corpos</span> </div> </a> <ul id="toc-Extensión_de_corpos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Véxase_tamén" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véxase_tamén"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Véxase tamén</span> </div> </a> <button aria-controls="toc-Véxase_tamén-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>Mostrar ou agochar a subsección "Véxase tamén"</span> </button> <ul id="toc-Véxase_tamén-sublist" class="vector-toc-list"> <li id="toc-Ligazóns_externas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ligazóns_externas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Ligazóns externas</span> </div> </a> <ul id="toc-Ligazóns_externas-sublist" class="vector-toc-list"> </ul> </li> </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="Contidos" 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="Mostrar ou agochar a táboa de contidos" > <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">Mostrar ou agochar a táboa de contidos</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">Corpo (álxebra)</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="Ir a un artigo noutra lingua. Dispoñible en 64 linguas" > <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-64" 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">64 linguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%82%D9%84_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="حقل (رياضيات) – árabe" lang="ar" hreflang="ar" data-title="حقل (رياضيات)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%AF%D0%BB%D0%B0%D0%BD_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Ялан (алгебра) – baxkir" lang="ba" hreflang="ba" data-title="Ялан (алгебра)" data-language-autonym="Башҡортса" data-language-local-name="baxkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Поле (алгебра) – belaruso" lang="be" hreflang="be" data-title="Поле (алгебра)" data-language-autonym="Беларуская" data-language-local-name="belaruso" 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%9F%D0%BE%D0%BB%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Поле (алгебра) – búlgaro" lang="bg" hreflang="bg" data-title="Поле (алгебра)" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="ক্ষেত্র (গণিত) – bengalí" lang="bn" hreflang="bn" data-title="ক্ষেত্র (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengalí" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Polje_(matematika)" title="Polje (matematika) – bosníaco" lang="bs" hreflang="bs" data-title="Polje (matematika)" data-language-autonym="Bosanski" data-language-local-name="bosníaco" 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/Cos_(matem%C3%A0tiques)" title="Cos (matemàtiques) – catalán" lang="ca" hreflang="ca" data-title="Cos (matemàtiques)" data-language-autonym="Català" data-language-local-name="catalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%DB%95%DB%8C%D8%AF%D8%A7%D9%86_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="مەیدان (ماتماتیک) – kurdo central" lang="ckb" hreflang="ckb" data-title="مەیدان (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="kurdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Komutativn%C3%AD_t%C4%9Bleso" title="Komutativní těleso – checo" lang="cs" hreflang="cs" data-title="Komutativní těleso" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D0%B9_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Уй (алгебра) – chuvaxo" lang="cv" hreflang="cv" data-title="Уй (алгебра)" data-language-autonym="Чӑвашла" data-language-local-name="chuvaxo" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Legeme_(algebra)" title="Legeme (algebra) – dinamarqués" lang="da" hreflang="da" data-title="Legeme (algebra)" data-language-autonym="Dansk" data-language-local-name="dinamarqués" 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/K%C3%B6rper_(Algebra)" title="Körper (Algebra) – alemán" lang="de" hreflang="de" data-title="Körper (Algebra)" data-language-autonym="Deutsch" data-language-local-name="alemán" 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%A3%CF%8E%CE%BC%CE%B1_(%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1)" title="Σώμα (άλγεβρα) – grego" lang="el" hreflang="el" data-title="Σώμα (άλγεβρα)" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="artigo bo"><a href="https://en.wikipedia.org/wiki/Field_(mathematics)" title="Field (mathematics) – inglés" lang="en" hreflang="en" data-title="Field (mathematics)" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kampo_(algebro)" title="Kampo (algebro) – esperanto" lang="eo" hreflang="eo" data-title="Kampo (algebro)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuerpo_(matem%C3%A1ticas)" title="Cuerpo (matemáticas) – español" lang="es" hreflang="es" data-title="Cuerpo (matemáticas)" data-language-autonym="Español" data-language-local-name="español" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Korpus_(matemaatika)" title="Korpus (matemaatika) – estoniano" lang="et" hreflang="et" data-title="Korpus (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estoniano" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Gorputz_(matematika)" title="Gorputz (matematika) – éuscaro" lang="eu" hreflang="eu" data-title="Gorputz (matematika)" data-language-autonym="Euskara" data-language-local-name="éuscaro" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="میدان (ریاضیات) – persa" lang="fa" hreflang="fa" data-title="میدان (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kunta_(matematiikka)" title="Kunta (matematiikka) – finés" lang="fi" hreflang="fi" data-title="Kunta (matematiikka)" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Korpus_(mat%C3%B5maatiga)" title="Korpus (matõmaatiga) – Võro" lang="vro" hreflang="vro" data-title="Korpus (matõmaatiga)" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target"><span>Võro</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Corps_commutatif" title="Corps commutatif – francés" lang="fr" hreflang="fr" data-title="Corps commutatif" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9imse_(matamaitic)" title="Réimse (matamaitic) – irlandés" lang="ga" hreflang="ga" data-title="Réimse (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irlandés" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%93%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="שדה (מבנה אלגברי) – hebreo" lang="he" hreflang="he" data-title="שדה (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4)" title="क्षेत्र (गणित) – hindi" lang="hi" hreflang="hi" data-title="क्षेत्र (गणित)" 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/Polje_(matematika)" title="Polje (matematika) – croata" lang="hr" hreflang="hr" data-title="Polje (matematika)" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Test_(algebra)" title="Test (algebra) – húngaro" lang="hu" hreflang="hu" data-title="Test (algebra)" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Corpore_(mathematica)" title="Corpore (mathematica) – interlingua" lang="ia" hreflang="ia" data-title="Corpore (mathematica)" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Lapangan_(matematika)" title="Lapangan (matematika) – indonesio" lang="id" hreflang="id" data-title="Lapangan (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Feldo_(algebro)" title="Feldo (algebro) – ido" lang="io" hreflang="io" data-title="Feldo (algebro)" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Campo_(matematica)" title="Campo (matematica) – italiano" lang="it" hreflang="it" data-title="Campo (matematica)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%8F%AF%E6%8F%9B%E4%BD%93" title="可換体 – xaponés" lang="ja" hreflang="ja" data-title="可換体" data-language-autonym="日本語" data-language-local-name="xaponés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학) – coreano" lang="ko" hreflang="ko" data-title="체 (수학)" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Corpus_(mathematica)" title="Corpus (mathematica) – latín" lang="la" hreflang="la" data-title="Corpus (mathematica)" data-language-autonym="Latina" data-language-local-name="latín" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Kierper_(Algeber)" title="Kierper (Algeber) – luxemburgués" lang="lb" hreflang="lb" data-title="Kierper (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburgués" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Camp_(matematega)" title="Camp (matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="Camp (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Lauks_(matem%C4%81tika)" title="Lauks (matemātika) – letón" lang="lv" hreflang="lv" data-title="Lauks (matemātika)" data-language-autonym="Latviešu" data-language-local-name="letón" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Medan_(matematik)" title="Medan (matematik) – malaio" lang="ms" hreflang="ms" data-title="Medan (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Lichaam_(Ned)_/_Veld_(Be)" title="Lichaam (Ned) / Veld (Be) – neerlandés" lang="nl" hreflang="nl" data-title="Lichaam (Ned) / Veld (Be)" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Kropp_i_matematikk" title="Kropp i matematikk – noruegués nynorsk" lang="nn" hreflang="nn" data-title="Kropp i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="noruegués nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Tallkropp" title="Tallkropp – noruegués bokmål" lang="nb" hreflang="nb" data-title="Tallkropp" data-language-autonym="Norsk bokmål" data-language-local-name="noruegués bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Cia%C5%82o_(matematyka)" title="Ciało (matematyka) – polaco" lang="pl" hreflang="pl" data-title="Ciało (matematyka)" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Camp_(matem%C3%A0tica)" title="Camp (matemàtica) – Piedmontese" lang="pms" hreflang="pms" data-title="Camp (matemàtica)" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática) – portugués" lang="pt" hreflang="pt" data-title="Corpo (matemática)" data-language-autonym="Português" data-language-local-name="portugués" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Corp_comutativ" title="Corp comutativ – romanés" lang="ro" hreflang="ro" data-title="Corp comutativ" data-language-autonym="Română" data-language-local-name="romanés" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Поле (алгебра) – ruso" lang="ru" hreflang="ru" data-title="Поле (алгебра)" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Campu_(matimatica)" title="Campu (matimatica) – siciliano" lang="scn" hreflang="scn" data-title="Campu (matimatica)" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Polje_(matematika)" title="Polje (matematika) – serbocroata" lang="sh" hreflang="sh" data-title="Polje (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Field_(mathematics)" title="Field (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Field (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Pole_(algebra)" title="Pole (algebra) – eslovaco" lang="sk" hreflang="sk" data-title="Pole (algebra)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Fusha_(matematik%C3%AB)" title="Fusha (matematikë) – albanés" lang="sq" hreflang="sq" data-title="Fusha (matematikë)" data-language-autonym="Shqip" data-language-local-name="albanés" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D1%99%D0%B5_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Поље (математика) – serbio" lang="sr" hreflang="sr" data-title="Поље (математика)" data-language-autonym="Српски / srpski" data-language-local-name="serbio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kropp_(algebra)" title="Kropp (algebra) – sueco" lang="sv" hreflang="sv" data-title="Kropp (algebra)" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%B3%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="களம் (கணிதம்) – támil" lang="ta" hreflang="ta" data-title="களம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="támil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9F%E0%B8%B5%E0%B8%A5%E0%B8%94%E0%B9%8C" title="ฟีลด์ – tailandés" lang="th" hreflang="th" data-title="ฟีลด์" data-language-autonym="ไทย" data-language-local-name="tailandés" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Cisim_(cebir)" title="Cisim (cebir) – turco" lang="tr" hreflang="tr" data-title="Cisim (cebir)" data-language-autonym="Türkçe" data-language-local-name="turco" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Поле (алгебра) – ucraíno" lang="uk" hreflang="uk" data-title="Поле (алгебра)" data-language-autonym="Українська" data-language-local-name="ucraíno" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86" title="میدان – urdú" lang="ur" hreflang="ur" data-title="میدان" data-language-autonym="اردو" data-language-local-name="urdú" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Tr%C6%B0%E1%BB%9Dng_(%C4%91%E1%BA%A1i_s%E1%BB%91)" title="Trường (đại số) – vietnamita" lang="vi" hreflang="vi" data-title="Trường (đại số)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%9F%9F%EF%BC%88%E6%95%B0%E5%AD%A6%EF%BC%89" title="域（数学） – chinés wu" lang="wuu" hreflang="wuu" data-title="域（数学）" data-language-autonym="吴语" data-language-local-name="chinés wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9F%9F_(%E6%95%B0%E5%AD%A6)" title="域 (数学) – chinés" lang="zh" hreflang="zh" data-title="域 (数学)" data-language-autonym="中文" data-language-local-name="chinés" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%9F%9F_(%E4%BB%A3%E6%95%B8)" title="域 (代數) – Literary Chinese" lang="lzh" hreflang="lzh" data-title="域 (代數)" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Th%C3%A9_(s%C3%B2%CD%98-ha%CC%8Dk)" title="Thé (sò͘-ha̍k) – Minnan" lang="nan" hreflang="nan" data-title="Thé (sò͘-ha̍k)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%AB%94_(%E6%95%B8%E5%AD%B8)" title="體 (數學) – cantonés" lang="yue" hreflang="yue" data-title="體 (數學)" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q190109#sitelinks-wikipedia" title="Editar as ligazóns interlingüísticas" class="wbc-editpage">Editar as ligazóns</a></span></div> </div> </div> </div> </header> <div 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data-event-name="pinnable-header.vector-appearance.unpin">agochar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Na Galipedia, a Wikipedia en galego.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="gl" dir="ltr"><p>Na <a href="/wiki/%C3%81lxebra_abstracta" title="Álxebra abstracta">álxebra abstracta</a>, un <b>corpo</b> é unha <a href="/wiki/Estrutura_alx%C3%A9brica" title="Estrutura alxébrica">estrutura alxébrica</a> na cal se dan as operacións de adición e multiplicación que ademais cumpren as propiedades <a href="/wiki/Propiedade_asociativa" class="mw-redirect" title="Propiedade asociativa">asociativa</a>, <a href="/wiki/Propiedade_conmutativa" class="mw-redirect" title="Propiedade conmutativa">conmutativa</a> e <a href="/wiki/Propiedade_distributiva" class="mw-redirect" title="Propiedade distributiva">distributiva</a>, e posúen un inverso aditivo e un inverso multiplicativo, que permiten efectuar as operacións de <a href="/wiki/Resta" class="mw-redirect" title="Resta">resta</a> e <a href="/wiki/Divisi%C3%B3n_(matem%C3%A1ticas)" title="División (matemáticas)">división</a> (excepto a división por cero). Estas propiedades xa son familiares da aritmética de números ordinarios. </p><p>Os corpos son obxectos importantes de estudo na álxebra posto que proporcionan a xeneralización apropiada de dominios de números tales como os conxuntos de <a href="/wiki/N%C3%BAmero_racional" title="Número racional">números racionais</a>, dos <a href="/wiki/N%C3%BAmero_real" title="Número real">números reais</a> ou dos <a href="/wiki/N%C3%BAmero_complexo" title="Número complexo">números complexos</a>. </p><p>O concepto dun corpo emprégase tamén para definir o concepto de <a href="/wiki/Espazo_vectorial" title="Espazo vectorial">espazo vectorial</a> e as transformacións nestes obxectos, dadas por <a href="/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas)">matrices</a>, obxectos que na <a href="/wiki/%C3%81lxebra_linear" class="mw-redirect" title="Álxebra linear">álxebra linear</a> os seus compoñentes poden ser elementos dun corpo arbitrario. A <a href="/wiki/Teor%C3%ADa_de_Galois" title="Teoría de Galois">teoría de Galois</a> estuda as relacións de simetría nas ecuacións alxébricas, desde a observación do comportamento das súas raíces e as extensións de corpos correspondentes e a súa relación cos automorfismos de corpos correspondentes. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definición"><span id="Definici.C3.B3n"></span>Definición</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=1" title="Editar a sección: «Definición»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=1" title="Editar o código fonte da sección: Definición"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Un corpo é un <a href="/w/index.php?title=Anel_de_divisi%C3%B3n&action=edit&redlink=1" class="new" title="Anel de división (a páxina aínda non existe)">anel de división</a> <a href="/wiki/Anel_conmutativo" title="Anel conmutativo">conmutativo</a>, é dicir, un anel conmutativo e <a href="/wiki/Anel_unitario" class="mw-redirect" title="Anel unitario">unitario</a> no que todo elemento distinto de cero é invertible respecto do produto. Polo tanto, un corpo é un conxunto <i>K</i> no que se definiron dúas <a href="/wiki/Operaci%C3%B3n_matem%C3%A1tica" class="mw-redirect" title="Operación matemática">operacións</a>, + e ·, chamadas <i>suma</i> e <i>multiplicación</i> respectivamente, que cumpren as seguintes propiedades: </p><p><b><i>K</i> é cerrado para a suma e a multiplicación</b> </p> <dl><dd>Para todo <i>a</i>, <i>b</i> en <i>K</i>, <i>a</i> + <i>b</i> e <i>a</i> * <i>b</i> pertencen a <i>K</i> (ou máis formalmente, + e * son <a href="/wiki/Operaci%C3%B3n_matem%C3%A1tica" class="mw-redirect" title="Operación matemática">operacións matemáticas</a> en <i>K</i>);</dd></dl> <p><b><a href="/wiki/Asociatividade" class="mw-redirect" title="Asociatividade">Asociatividade</a> da suma e a multiplicación</b> </p> <dl><dd>Para todo <i>a</i>, <i>b</i>, <i>c</i> en <i>K</i>, <i>a</i> + (<i>b</i> + <i>c</i>) = (<i>a</i> + <i>b</i>) + <i>c</i> e <i>a</i> * (<i>b</i> * <i>c</i>) = (<i>a</i> * <i>b</i>) * <i>c</i>.</dd></dl> <p><b><a href="/wiki/Conmutatividade" title="Conmutatividade">Conmutatividade</a> da suma e a multiplicación</b> </p> <dl><dd>Para todo <i>a</i>, <i>b</i> en <i>K</i>, <i>a</i> + <i>b</i> = <i>b</i> + <i>a</i> e <i>a</i> * <i>b</i> = <i>b</i> * <i>a</i>.</dd></dl> <p><b>Existencia dun <a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a> para a suma e a multiplicación</b> </p> <dl><dd>Existe un elemento 0 en <i>K</i>, tal que para todo <i>a</i> en <i>K</i>, <i>a</i> + 0 = <i>a</i>.</dd> <dd>Existe un elemento 1 en <i>K</i> diferente a 0, tal que para todo <i>a</i> en <i>K</i>, <i>a</i> * 1 = <i>a</i>.</dd></dl> <p><b>Existencia de <a href="/w/index.php?title=Elemento_oposto&action=edit&redlink=1" class="new" title="Elemento oposto (a páxina aínda non existe)">elemento oposto</a> e de inversos:</b> </p> <dl><dd>Para cada <i>a</i> en <i>K</i>, existe un elemento -<i>a</i> en <i>K</i>, tal que <i>a</i> + (- <i>a</i>) = 0.</dd> <dd>Para cada <i>a</i> ≠ 0 en <i>K</i>, existe un elemento <i>a</i> <sup>-1</sup> en <i>K</i>, tal que <i>a</i> * <i>a</i><sup>-1</sup> = 1.</dd></dl> <p><b><a href="/wiki/Distributividade" title="Distributividade">Distributividade</a> da multiplicación respecto da suma</b> </p> <dl><dd>Para todo <i>a</i>, <i>b</i>, <i>c</i>, en <i>K</i>, <i>a</i> * (<i>b</i> + <i>c</i>) = (<i>a</i> * <i>b</i>) + (<i>a</i> * <i>c</i>).</dd></dl> <p>O requisito a ≠ 0 asegura que o conxunto que contén só un cero non sexa un corpo, e de paso elimina a posibilidade de que no corpo existan <a href="/wiki/Divisor_de_cero" title="Divisor de cero">divisores de cero</a> distintos de 0, o que o converte tamén nun <a href="/w/index.php?title=Dominio_de_integridade&action=edit&redlink=1" class="new" title="Dominio de integridade (a páxina aínda non existe)">dominio de integridade</a>. Directamente dos axiomas, pódese demostrar que (<i>K</i>, +) e (<i>K</i> - { 0 }, *) son <a href="/wiki/Grupo_abeliano" title="Grupo abeliano">grupos conmutativos</a> e que polo tanto (véxase a <a href="/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos">teoría de grupos</a>) o <a href="/w/index.php?title=Elemento_oposto&action=edit&redlink=1" class="new" title="Elemento oposto (a páxina aínda non existe)">oposto</a> -<i>a</i> e o <a href="/wiki/Inverso_multiplicativo" title="Inverso multiplicativo">inverso</a> <i>a</i><sup>−1</sup> son determinados unicamente por <i>a</i>. Ademais, o inverso dun produto é igual ao produto dos inversos: </p> <dl><dd>(<i>a*b</i>)<sup>-1</sup> = <i>a</i><sup>-1</sup> * <i>b</i><sup>-1</sup></dd></dl> <p>con tal que <i>a</i> e <i>b</i> sexan diferentes de cero. Outras regras útiles son: </p> <dl><dd>-<i>a</i> = (-1) * <i>a</i></dd></dl> <p>e máis xeralmente </p> <dl><dd>- (<i>a</i> * <i>b</i>) = (-<i>a</i>) * <i>b</i> = <i>a</i> * (-<i>b</i>)</dd></dl> <p>así como </p> <dl><dd><i>a</i> * 0 = 0,</dd></dl> <p>todas regras familiares da <a href="/wiki/Aritm%C3%A9tica" title="Aritmética">aritmética</a> elemental. </p> <div class="mw-heading mw-heading2"><h2 id="Exemplos_de_corpos">Exemplos de corpos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=2" title="Editar a sección: «Exemplos de corpos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=2" title="Editar o código fonte da sección: Exemplos de corpos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Racionais_e_alxébricos"><span id="Racionais_e_alx.C3.A9bricos"></span>Racionais e alxébricos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=3" title="Editar a sección: «Racionais e alxébricos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=3" title="Editar o código fonte da sección: Racionais e alxébricos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os <a href="/wiki/N%C3%BAmeros_racionais" class="mw-redirect" title="Números racionais">números racionais</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} =\{{a \over b}|a,b\in \mathbb {Z} ,b\neq 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mi>b</mi> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} =\{{a \over b}|a,b\in \mathbb {Z} ,b\neq 0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccbed3aaf76a9de118ee95b175adae6a1a6f61e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.889ex; height:4.843ex;" alt="{\displaystyle \mathbb {Q} =\{{a \over b}|a,b\in \mathbb {Z} ,b\neq 0\}}"></span> onde está incluído o conxunto <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> dos <a href="/wiki/N%C3%BAmeros_enteiros" class="mw-redirect" title="Números enteiros">números enteiros</a>, forman un corpo. </p><p>Os números complexos conteñen o corpo de <a href="/wiki/N%C3%BAmero_alx%C3%A9brico" title="Número alxébrico">números alxébricos</a>, a <a href="/w/index.php?title=Clausura_alx%C3%A9brica&action=edit&redlink=1" class="new" title="Clausura alxébrica (a páxina aínda non existe)">clausura alxébrica</a> de <b>Q</b>. </p> <div class="mw-heading mw-heading3"><h3 id="Números_reais,_complexos_e_p-ádicos"><span id="N.C3.BAmeros_reais.2C_complexos_e_p-.C3.A1dicos"></span>Números reais, complexos e <i>p</i>-ádicos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=4" title="Editar a sección: «Números reais, complexos e p-ádicos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=4" title="Editar o código fonte da sección: Números reais, complexos e p-ádicos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Os <a href="/wiki/N%C3%BAmeros_reais" class="mw-redirect" title="Números reais">números reais</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> coas operacións usuais forman un corpo. </p><p>Os <a href="/w/index.php?title=N%C3%BAmero_hiperreal&action=edit&redlink=1" class="new" title="Número hiperreal (a páxina aínda non existe)">números hiperreais</a> forman un corpo que contén os reais máis os números infinitesimais e infinitos. Os <a href="/w/index.php?title=N%C3%BAmero_surreal&action=edit&redlink=1" class="new" title="Número surreal (a páxina aínda non existe)">números surreais</a> forman un corpo que contén os reais, a excepción do feito de que son unha clase propia, non un conxunto. O conxunto de todos os números surreais coa condición de seren menor que un certo <a href="/w/index.php?title=Cardinal_inaccesible&action=edit&redlink=1" class="new" title="Cardinal inaccesible (a páxina aínda non existe)">cardinal inaccesible</a> é un corpo. </p><p>Os números reais conteñen varios subcorpos interesantes: os números reais alxébricos, os <a href="/w/index.php?title=N%C3%BAmero_computable&action=edit&redlink=1" class="new" title="Número computable (a páxina aínda non existe)">números computables</a>, e os <a href="/w/index.php?title=N%C3%BAmero_definible&action=edit&redlink=1" class="new" title="Número definible (a páxina aínda non existe)">números definibles</a>. </p><p>Os <a href="/wiki/N%C3%BAmeros_complexos" class="mw-redirect" title="Números complexos">números complexos</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> consisten en expresións do tipo </p> <dl><dd><i>a</i> + <i>b</i>i</dd></dl> <p>onde i é a <a href="/wiki/Unidade_imaxinaria" title="Unidade imaxinaria">unidade imaxinaria</a>, <i>i.e.</i>, un número (non real) que satisfai i<sup>2</sup> = −1. A adición e multiplicación dos números reais son definidos de tal maneira para que todos os axiomas do corpo se cumpren para <b>C</b>. Por exemplo, a lei distributiva cumpre </p> <dl><dd>(<i>a</i> + <i>b</i>i)·(<i>c</i> + <i>d</i>i) = <i>ac</i> + <i>bc</i>i + <i>ad</i>i + <i>bd</i>i<sup>2</sup>, que é igual a <i>ac</i>−<i>bd</i> + (<i>bc</i> + <i>ad</i>)i.</dd></dl> <p>Os números racionais pódense ampliar aos corpos de <a href="/wiki/N%C3%BAmero_p-%C3%A1dico" title="Número p-ádico">números p-ádicos</a> para cada número primo <i>p</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Corpos_finitos">Corpos finitos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=5" title="Editar a sección: «Corpos finitos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=5" title="Editar o código fonte da sección: Corpos finitos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>O corpo máis pequeno ten só dous elementos: 0 e 1. É denotado por <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 {F} }_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {F} }_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ab38e4f9b846a1b8478d494356bff0de055bac5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.475ex; height:2.509ex;" alt="{\displaystyle {\mathbb {F} }_{2}}"></span> o <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} }_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Z} }_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf936518529371afb91aa66a18eee5e3e29fa00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.605ex; height:2.509ex;" alt="{\displaystyle {\mathbb {Z} }_{2}}"></span> e pode ás veces ser visto nas dúas táboas que seguen: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{array}{|c|c|c|}\hline +&\mathbf {0} &\mathbf {1} \\\hline \mathbf {0} &0&1\\\mathbf {1} &1&0\\\hline \end{array}}\quad {\begin{array}{|c|c|c|}\hline \cdot &\mathbf {0} &\mathbf {1} \\\hline \mathbf {0} &0&0\\\mathbf {1} &0&1\\\hline \end{array}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid" frame="solid"> <mtr> <mtd> <mo>+</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em" rowlines="solid none" columnlines="solid solid" frame="solid"> <mtr> <mtd> <mo>⋅</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{array}{|c|c|c|}\hline +&\mathbf {0} &\mathbf {1} \\\hline \mathbf {0} &0&1\\\mathbf {1} &1&0\\\hline \end{array}}\quad {\begin{array}{|c|c|c|}\hline \cdot &\mathbf {0} &\mathbf {1} \\\hline \mathbf {0} &0&0\\\mathbf {1} &0&1\\\hline \end{array}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4790305ea0322f931f114354e1ad21eb79dc2c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:25.323ex; height:10.176ex;" alt="{\displaystyle {\begin{array}{|c|c|c|}\hline +&\mathbf {0} &\mathbf {1} \\\hline \mathbf {0} &0&1\\\mathbf {1} &1&0\\\hline \end{array}}\quad {\begin{array}{|c|c|c|}\hline \cdot &\mathbf {0} &\mathbf {1} \\\hline \mathbf {0} &0&0\\\mathbf {1} &0&1\\\hline \end{array}}}"></span></dd></dl> <p>Ten aplicacións importantes na <a href="/wiki/Inform%C3%A1tica" title="Informática">informática</a>, especialmente na <a href="/wiki/Criptograf%C3%ADa" title="Criptografía">criptografía</a> e na <a href="/w/index.php?title=Teor%C3%ADa_da_codificaci%C3%B3n&action=edit&redlink=1" class="new" title="Teoría da codificación (a páxina aínda non existe)">teoría da codificación</a>. </p><p>Máis xeralmente, para un <a href="/wiki/N%C3%BAmero_primo" title="Número primo">número primo</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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>, o conxunto dos números enteiros <a href="/wiki/M%C3%B3dulo_(%C3%A1lxebra)" title="Módulo (álxebra)">módulo</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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> é un corpo finito cos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> elementos: isto pode escribirse como <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {Z} }_{p}=\{0,1,...,p-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>p</mi> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbb {Z} }_{p}=\{0,1,...,p-1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558b67823c669df7ef3e93503caca78074c57e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.734ex; height:3.009ex;" alt="{\displaystyle {\mathbb {Z} }_{p}=\{0,1,...,p-1\}}"></span> onde as operacións son definidas realizando a operación en <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>, dividindo por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> e tomando o resto, ver <a href="/wiki/Aritm%C3%A9tica_modular" title="Aritmética modular">aritmética modular</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Corpos_de_funcións"><span id="Corpos_de_funci.C3.B3ns"></span>Corpos de funcións</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=6" title="Editar a sección: «Corpos de funcións»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=6" title="Editar o código fonte da sección: Corpos de funcións"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para un corpo dado <i>K</i>, o conxunto <i>K</i>(<i>X</i>) de funcións racionais na variable <i>X</i> con coeficientes en <i>K</i> é un corpo; isto defínese como o conxunto de cocientes de <a href="/wiki/Polinomio" title="Polinomio">polinomios</a> con coeficientes en <i>K</i>. </p><p>Se <i>K</i> é corpo, e <i>p</i>(<i>X</i>) é un <a href="/w/index.php?title=Polinomio_irreducible&action=edit&redlink=1" class="new" title="Polinomio irreducible (a páxina aínda non existe)">polinomio irreducible</a> nun anel de polinomios <i>F</i>[<i>X</i>], entón o cociente <i>F</i>[<i>X</i>]/<<i>p</i>(<i>X</i>)> é un corpo cun subcorpo isomorfo a <i>K</i>. Por exemplo, <b>R</b>[<i>X</i>]/(<i>X</i><sup>2</sup>+1) é un corpo (de feito, é isomorfo con respecto ao corpo dos números complexos). </p><p>Cando <i>K</i> é un corpo, o conxunto <i>K</i>[[X]] de <a href="/w/index.php?title=Serie_de_Laurent&action=edit&redlink=1" class="new" title="Serie de Laurent (a páxina aínda non existe)">series formais de Laurent</a> sobre <i>K</i> é un corpo. </p><p>Se <i>V</i> é unha <a href="/wiki/Variedade_alx%C3%A9brica" title="Variedade alxébrica">variedade alxébrica</a> sobre <i>K</i>, entón as funcións racionais <i>V</i> → <i>K</i> forman un corpo, o corpo de funcións <i>V</i>. Se <i>S</i> é unha <a href="/wiki/Superficie_de_Riemann" title="Superficie de Riemann">superficie de Riemann</a>, entón as funcións <a href="/wiki/Funci%C3%B3n_meromorfa" title="Función meromorfa">meromorfas</a> de <i>S</i> → <b>C</b> forman un corpo. </p> <div class="mw-heading mw-heading3"><h3 id="Ultrafiltros">Ultrafiltros</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=7" title="Editar a sección: «Ultrafiltros»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=7" title="Editar o código fonte da sección: Ultrafiltros"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <i>I</i> é un conxunto de índices, <i>U</i> é un <a href="/w/index.php?title=Ultrafiltro&action=edit&redlink=1" class="new" title="Ultrafiltro (a páxina aínda non existe)">ultrafiltro</a> sobre <i>I</i>, e <i>K</i><sub><i>i</i></sub> é un corpo para cada <i>i</i> en <i>I</i>, o <a href="/w/index.php?title=Ultraproduto&action=edit&redlink=1" class="new" title="Ultraproduto (a páxina aínda non existe)">ultraproduto</a> de <i>K</i><sub><i>i</i></sub> (usando <i>U</i>) é un corpo. </p> <div class="mw-heading mw-heading3"><h3 id="Subcorpos">Subcorpos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=8" title="Editar a sección: «Subcorpos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=8" title="Editar o código fonte da sección: Subcorpos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sexan <i>E</i> e <i>K</i> dous corpos con <i>E</i> un subcorpo de <i>K</i> (é dicir, un subconxunto de <i>K</i> que contén 0 e 1, cerrado baixo as operacións + e * de <i>K</i> e coas súas propias operacións definidas por restrición). Sexa <i>x</i> un elemento de <i>K</i> non en <i>E</i>. Entón <i>E</i>(<i>x</i>) defínese como o subcorpo máis pequeno de <i>K</i> que contén a <i>E</i> e a <i>x</i>. Por exemplo, <b>Q</b>(<i>i</i>) é o subcorpo dos números complexos <b>C</b> que consisten en todos os números da forma <i>a+bi</i> onde <i>a</i> e <i>b</i> son números racionais. </p> <div class="mw-heading mw-heading2"><h2 id="Algúns_teoremas_iniciais"><span id="Alg.C3.BAns_teoremas_iniciais"></span>Algúns teoremas iniciais</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=9" title="Editar a sección: «Algúns teoremas iniciais»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=9" title="Editar o código fonte da sección: Algúns teoremas iniciais"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>O conxunto de elementos diferentes de cero dun corpo <i>K</i> (denotado tipicamente por <i>K</i><sup>×</sup>) é un grupo abeliano baixo multiplicación. Cada subgrupo finito de <i>K</i><sup>×</sup> é <a href="/wiki/Grupo_c%C3%ADclico" title="Grupo cíclico">cíclico</a>.</li> <li>A <a href="/w/index.php?title=Caracter%C3%ADstica_(%C3%A1lxebra)&action=edit&redlink=1" class="new" title="Característica (álxebra) (a páxina aínda non existe)">característica</a> de calquera corpo é cero ou un <a href="/wiki/N%C3%BAmero_primo" title="Número primo">número primo</a>. A característica defínese como o número enteiro positivo máis pequeno <i>n</i> tal que <i>n</i>·1 = 0, ou cero se non existe tal <i>n</i>; aquí <i>n</i>·1 significa <i>n</i> sumandos 1 + 1 + 1 +... + 1).</li> <li>Se <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>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe3606e1cf4480eb39e5ddcd46d4dae2067c0b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.33ex; height:2.509ex;" alt="{\displaystyle q>1}"></span> é unha potencia dun <a href="/wiki/N%C3%BAmero_primo" title="Número primo">número primo</a>, entón existe (salvo isomorfismo) exactamente un <a href="/w/index.php?title=Corpo_finito&action=edit&redlink=1" class="new" title="Corpo finito (a páxina aínda non existe)">corpo finito</a> con <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> elementos. Ademais, estes son os únicos corpos finitos posibles.</li> <li>Como anel, un corpo no ten ningún <a href="/w/index.php?title=Ideal_dun_anel&action=edit&redlink=1" class="new" title="Ideal dun anel (a páxina aínda non existe)">ideal</a> excepto {0} e el mesmo.</li> <li>Todo anel de división finito é un corpo (<a href="/w/index.php?title=Teorema_de_Wedderburn&action=edit&redlink=1" class="new" title="Teorema de Wedderburn (a páxina aínda non existe)">teorema de Wedderburn</a>).</li> <li>Para cada corpo <i>K</i>, existe (salvo isomorfismo) un corpo único G que contén a <i>K</i>, é <a href="/w/index.php?title=Elemento_alx%C3%A9brico&action=edit&redlink=1" class="new" title="Elemento alxébrico (a páxina aínda non existe)">alxébrico</a> sobre <i>K</i>, e é alxebricamente pechado. <i>G</i> denomínase <a href="/w/index.php?title=Clausura_alx%C3%A9brica&action=edit&redlink=1" class="new" title="Clausura alxébrica (a páxina aínda non existe)">clausura alxébrica</a> de <i>K</i>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Construcións_de_corpos"><span id="Construci.C3.B3ns_de_corpos"></span>Construcións de corpos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=10" title="Editar a sección: «Construcións de corpos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=10" title="Editar o código fonte da sección: Construcións de corpos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Subcorpos_e_ideais">Subcorpos e ideais</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=11" title="Editar a sección: «Subcorpos e ideais»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=11" title="Editar o código fonte da sección: Subcorpos e ideais"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se un subconxunto <i>E</i> dun corpo (<i>K</i>,+,*) xunto coas operacións *, + restrinxido a <i>E</i> é en si mesmo un corpo, entón denominase subcorpo de <i>K</i>. Tal subcorpo ten os mesmos 0 e 1 que <i>K</i>. </p><p>Sexa <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,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,+,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d474a4df1822902bbe09dab84bb9872f0019d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.398ex; height:2.843ex;" alt="{\displaystyle (K,+,\cdot )}"></span> un corpo, e <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 E\subset K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊂</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subset K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd383f94b2578aea88843b9be2ff158801f52032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.94ex; height:2.176ex;" alt="{\displaystyle E\subset K}"></span>. Dise que <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 \ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01358ea1a9419bfbfbaab2467c6db81c1a6b1fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.356ex; height:2.176ex;" alt="{\displaystyle \ E}"></span> é subcorpo de <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183b560070c7edf40847c6cd47918dc20eddd334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:2.176ex;" alt="{\displaystyle \ K}"></span> ou que <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183b560070c7edf40847c6cd47918dc20eddd334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:2.176ex;" alt="{\displaystyle \ K}"></span> é <a href="/w/index.php?title=Extensi%C3%B3n_de_corpo&action=edit&redlink=1" class="new" title="Extensión de corpo (a páxina aínda non existe)">extensión</a> de <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 \ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01358ea1a9419bfbfbaab2467c6db81c1a6b1fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.356ex; height:2.176ex;" alt="{\displaystyle \ E}"></span> se se cumpre que <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 (E,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E,+,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fccf8d1695d40ee2eb19abb243607e754306f98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.108ex; height:2.843ex;" alt="{\displaystyle (E,+,\cdot )}"></span> é un corpo cando as operacións <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 \ (+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe036aa8cf8bcc151462e325416b546999c11a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.198ex; height:2.843ex;" alt="{\displaystyle \ (+)}"></span> e <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 (\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c2b509ce21093c430b9c0849fa1aef7f0f1d24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.843ex;" alt="{\displaystyle (\cdot )}"></span> se restrinxen 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 \ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01358ea1a9419bfbfbaab2467c6db81c1a6b1fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.356ex; height:2.176ex;" alt="{\displaystyle \ E}"></span>. En particular, <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 \ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01358ea1a9419bfbfbaab2467c6db81c1a6b1fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.356ex; height:2.176ex;" alt="{\displaystyle \ E}"></span> será entón un <a href="/wiki/Anel_(matem%C3%A1ticas)" class="mw-redirect" title="Anel (matemáticas)">subanel</a> de <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,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K,+,\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d474a4df1822902bbe09dab84bb9872f0019d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.398ex; height:2.843ex;" alt="{\displaystyle (K,+,\cdot )}"></span>. Tense logo que <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 \ (E,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (E,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674d67dff8c96038fb7dd9f1b1efde7b5a7af3a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.008ex; height:2.843ex;" alt="{\displaystyle \ (E,+)}"></span> e <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 (E\setminus \{0\},\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>E</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E\setminus \{0\},\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c92134cb69002e6fdb031364629f82de9ec25edd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.948ex; height:2.843ex;" alt="{\displaystyle (E\setminus \{0\},\cdot )}"></span> son <a href="/wiki/Grupo_(matem%C3%A1ticas)" title="Grupo (matemáticas)">subgrupos</a> respectivos dos <a href="/wiki/Grupo_abeliano" title="Grupo abeliano">grupos abelianos</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 \ (K,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (K,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6e09f52e98fbda0c66f762a1a095be3896f8a75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.298ex; height:2.843ex;" alt="{\displaystyle \ (K,+)}"></span> e <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\setminus \{0\},\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>K</mi> <mo class="MJX-variant">∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (K\setminus \{0\},\cdot )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e47aa90e60a65ec3b03b46c9e7cfd8b1487404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.238ex; height:2.843ex;" alt="{\displaystyle (K\setminus \{0\},\cdot )}"></span>. </p><p>Como todo corpo é un anel, poderíamos preguntarnos pola forma que teñen os seus <a href="/w/index.php?title=Ideal_dun_anel&action=edit&redlink=1" class="new" title="Ideal dun anel (a páxina aínda non existe)">ideais</a>. Para empezar, como todo corpo é <a href="/wiki/Anel_conmutativo" title="Anel conmutativo">anel conmutativo</a>, todo ideal pola esquerda é ideal (bilátero) e todo ideal pola dereita é tamén ideal (bilátero). Así pois, só hai que estudar os ideais do corpo. </p><p>Se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e267f7ee4befa4ac2fabab76d0127a1adbc5eb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.752ex; height:2.176ex;" alt="{\displaystyle \ I}"></span> é ideal do corpo <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183b560070c7edf40847c6cd47918dc20eddd334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:2.176ex;" alt="{\displaystyle \ K}"></span>, entón todo elemento non nulo <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\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d838bfcfb39f7a33ffe31cd1c2a989b8ca3f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.136ex; height:2.176ex;" alt="{\displaystyle a\in K}"></span> terá inverso, <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^{-1}\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7615bf64892c54bab0cddfa8fc955c29fb5aeccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.469ex; height:2.676ex;" alt="{\displaystyle a^{-1}\in K}"></span>, logo <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14b4bc0b189c2d21060cd9ed1516a1f7e707c85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.81ex; height:1.676ex;" alt="{\displaystyle \ a}"></span> é unha unidade de <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/183b560070c7edf40847c6cd47918dc20eddd334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.647ex; height:2.176ex;" alt="{\displaystyle \ K}"></span> [isto é, <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\in U(K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in U(K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6724debd02faa73b27d01625aeb5248709621d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.728ex; height:2.843ex;" alt="{\displaystyle a\in U(K)}"></span>], e terase que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\cap U(K)\neq \varnothing }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>∩</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mi class="MJX-variant">∅</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\cap U(K)\neq \varnothing }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d62603176c0b9b1e251a2f217134ee52f88e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.319ex; height:2.843ex;" alt="{\displaystyle I\cap U(K)\neq \varnothing }"></span>, é dicir, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ I=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>I</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ I=R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dec391fb25c54d8c3f2fd370bcbdf20e8fe7989" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.615ex; height:2.176ex;" alt="{\displaystyle \ I=R}"></span>. Desta forma, os únicos ideais dun corpo son o propio corpo e o ideal nulo. </p> <div class="mw-heading mw-heading3"><h3 id="Corpo_de_fraccións"><span id="Corpo_de_fracci.C3.B3ns"></span>Corpo de fraccións</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=12" title="Editar a sección: «Corpo de fraccións»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=12" title="Editar o código fonte da sección: Corpo de fraccións"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dado un corpo K, o corpo polinómico <i>K</i>(<i>X</i>) é o <a href="/w/index.php?title=Corpo_de_fracci%C3%B3ns&action=edit&redlink=1" class="new" title="Corpo de fraccións (a páxina aínda non existe)">corpo de fraccións</a> de <a href="/wiki/Polinomio" title="Polinomio">polinomios</a> en <i>X</i> con coeficientes en <i>K</i>, é dicir, os seus elementos son <a href="/wiki/Funci%C3%B3n_racional" title="Función racional">funcións racionais</a> con coeficientes en K. </p> <div class="mw-heading mw-heading3"><h3 id="Extensión_de_corpos"><span id="Extensi.C3.B3n_de_corpos"></span>Extensión de corpos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=13" title="Editar a sección: «Extensión de corpos»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=13" title="Editar o código fonte da sección: Extensión de corpos"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unha <a href="/w/index.php?title=Extensi%C3%B3n_alx%C3%A9brica&action=edit&redlink=1" class="new" title="Extensión alxébrica (a páxina aínda non existe)">extensión alxébrica</a> dun corpo <i>K</i> é o corpo máis pequeno que contén a <i>K</i> e unha raíz dun polinomio irreducible <i>p</i>(<i>X</i>) en <i>K</i> [<i>X</i>]. Alternativamente, é idéntico ao <a href="/w/index.php?title=Anel_factor&action=edit&redlink=1" class="new" title="Anel factor (a páxina aínda non existe)">anel factor</a> <i>K</i> [<i>X</i>]/(<i>p</i>(<i>X</i>)), onde (<i>p</i>(<i>X</i>)) é o ideal xerado por <i>p</i>(<i>X</i>). </p> <div class="mw-heading mw-heading2"><h2 id="Véxase_tamén"><span id="V.C3.A9xase_tam.C3.A9n"></span>Véxase tamén</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=14" title="Editar a sección: «Véxase tamén»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=14" title="Editar o código fonte da sección: Véxase tamén"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Ligazóns_externas"><span id="Ligaz.C3.B3ns_externas"></span>Ligazóns externas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&veaction=edit&section=15" title="Editar a sección: «Ligazóns externas»" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Corpo_(%C3%A1lxebra)&action=edit&section=15" title="Editar o código fonte da sección: Ligazóns externas"><span>editar a fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/FieldAxioms.html">Field</a></i>, en <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com">Math World</a> <i>(en <a href="/wiki/Lingua_inglesa" title="Lingua inglesa">inglés</a>)</i></li></ul> <div role="navigation" class="navbox" aria-labelledby="Control_de_autoridades" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="Control_de_autoridades" scope="row" class="navbox-group" style="width:1%;width: 12%; 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