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class="vector-toc-link" href="#Généralisation"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Généralisation</span> </div> </a> <ul id="toc-Généralisation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algèbres_associatives,_algèbres_commutatives_et_algèbres_unifères" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algèbres_associatives,_algèbres_commutatives_et_algèbres_unifères"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Algèbres associatives, algèbres commutatives et algèbres unifères</span> </div> </a> <ul id="toc-Algèbres_associatives,_algèbres_commutatives_et_algèbres_unifères-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bases_et_tables_de_multiplication_d'une_algèbre_sur_un_corps" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bases_et_tables_de_multiplication_d'une_algèbre_sur_un_corps"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bases et tables de multiplication d'une algèbre sur un corps</span> </div> </a> <ul id="toc-Bases_et_tables_de_multiplication_d'une_algèbre_sur_un_corps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemple_d'algèbre_de_dimension_infinie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemple_d'algèbre_de_dimension_infinie"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Exemple d'algèbre de dimension infinie</span> </div> </a> <ul id="toc-Exemple_d'algèbre_de_dimension_infinie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemples_d'algèbres_de_dimension_finie" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemples_d'algèbres_de_dimension_finie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Exemples d'algèbres de dimension finie</span> </div> </a> <button aria-controls="toc-Exemples_d'algèbres_de_dimension_finie-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>Afficher / masquer la sous-section Exemples d'algèbres de dimension finie</span> </button> <ul id="toc-Exemples_d'algèbres_de_dimension_finie-sublist" class="vector-toc-list"> <li id="toc-Algèbres_associatives_et_commutatives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algèbres_associatives_et_commutatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Algèbres associatives et commutatives</span> </div> </a> <ul id="toc-Algèbres_associatives_et_commutatives-sublist" class="vector-toc-list"> <li id="toc-Nombres_complexes" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Nombres_complexes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.1</span> <span>Nombres complexes</span> </div> </a> <ul id="toc-Nombres_complexes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Corps_finis" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Corps_finis"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.2</span> <span>Corps finis</span> </div> </a> <ul id="toc-Corps_finis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algèbres_quadratiques" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Algèbres_quadratiques"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1.3</span> <span>Algèbres quadratiques</span> </div> </a> <ul id="toc-Algèbres_quadratiques-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algèbres_associatives_et_non_commutatives" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algèbres_associatives_et_non_commutatives"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Algèbres associatives et non commutatives</span> </div> </a> <ul id="toc-Algèbres_associatives_et_non_commutatives-sublist" class="vector-toc-list"> <li id="toc-Matrices_carrées" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Matrices_carrées"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Matrices carrées</span> </div> </a> <ul id="toc-Matrices_carrées-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quaternions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quaternions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>Quaternions</span> </div> </a> <ul id="toc-Quaternions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biquaternions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Biquaternions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.3</span> <span>Biquaternions</span> </div> </a> <ul id="toc-Biquaternions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algèbre_unifère_non_associative" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algèbre_unifère_non_associative"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Algèbre unifère non associative</span> </div> </a> <ul id="toc-Algèbre_unifère_non_associative-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algèbres_non_associatives_et_non_unifères" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algèbres_non_associatives_et_non_unifères"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Algèbres non associatives et non unifères</span> </div> </a> <ul id="toc-Algèbres_non_associatives_et_non_unifères-sublist" class="vector-toc-list"> <li id="toc-Produit_vectoriel" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Produit_vectoriel"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.1</span> <span>Produit vectoriel</span> </div> </a> <ul id="toc-Produit_vectoriel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Crochet_de_Lie" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Crochet_de_Lie"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.2</span> <span>Crochet de Lie</span> </div> </a> <ul id="toc-Crochet_de_Lie-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Contre-exemple" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Contre-exemple"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Contre-exemple</span> </div> </a> <ul id="toc-Contre-exemple-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voir_aussi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Voir_aussi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Voir aussi</span> </div> </a> <ul id="toc-Voir_aussi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_et_références" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_et_références"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes et références</span> </div> </a> <ul id="toc-Notes_et_références-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algèbre sur un corps</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="Aller à un article dans une autre langue. Disponible en 27 langues." > <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-27" 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">27 langues</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%AC%D8%A8%D8%B1_%D8%B9%D9%84%D9%89_%D8%AD%D9%82%D9%84" title="جبر على حقل – arabe" lang="ar" hreflang="ar" data-title="جبر على حقل" data-language-autonym="العربية" data-language-local-name="arabe" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BD%D0%B0%D0%B4_%D0%BF%D0%BE%D0%BB%D0%B5" title="Алгебра над поле – bulgare" lang="bg" hreflang="bg" data-title="Алгебра над поле" data-language-autonym="Български" data-language-local-name="bulgare" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_sobre_un_cos" title="Àlgebra sobre un cos – catalan" lang="ca" hreflang="ca" data-title="Àlgebra sobre un cos" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algebra_(struktura)" title="Algebra (struktura) – tchèque" lang="cs" hreflang="cs" data-title="Algebra (struktura)" data-language-autonym="Čeština" data-language-local-name="tchèque" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_(%D1%83%D0%B9_%C3%A7%D0%B8%D0%B9%C4%95%D0%BD)" title="Алгебра (уй çийĕн) – tchouvache" lang="cv" hreflang="cv" data-title="Алгебра (уй çийĕн)" data-language-autonym="Чӑвашла" data-language-local-name="tchouvache" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper – allemand" lang="de" hreflang="de" data-title="Algebra über einem Körper" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Algebra_over_a_field" title="Algebra over a field – anglais" lang="en" hreflang="en" data-title="Algebra over a field" data-language-autonym="English" data-language-local-name="anglais" 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/Al%C4%9Debro" title="Alĝebro – espéranto" lang="eo" hreflang="eo" data-title="Alĝebro" data-language-autonym="Esperanto" data-language-local-name="espéranto" 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/%C3%81lgebra_sobre_un_cuerpo" title="Álgebra sobre un cuerpo – espagnol" lang="es" hreflang="es" data-title="Álgebra sobre un cuerpo" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%B1%D9%88%DB%8C_%DB%8C%DA%A9_%D9%85%DB%8C%D8%AF%D8%A7%D9%86" title="جبر روی یک میدان – persan" lang="fa" hreflang="fa" data-title="جبر روی یک میدان" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_sobre_un_corpo" title="Álxebra sobre un corpo – galicien" lang="gl" hreflang="gl" data-title="Álxebra sobre un corpo" data-language-autonym="Galego" data-language-local-name="galicien" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algebra_super_un_corpore" title="Algebra super un corpore – interlingua" lang="ia" hreflang="ia" data-title="Algebra super un corpore" 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/Aljabar_atas_medan" title="Aljabar atas medan – indonésien" lang="id" hreflang="id" data-title="Aljabar atas medan" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésien" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algebra_su_campo" title="Algebra su campo – italien" lang="it" hreflang="it" data-title="Algebra su campo" data-language-autonym="Italiano" data-language-local-name="italien" 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/%E4%BD%93%E4%B8%8A%E3%81%AE%E5%A4%9A%E5%85%83%E7%92%B0" title="体上の多元環 – japonais" lang="ja" hreflang="ja" data-title="体上の多元環" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra_(structuur)" title="Algebra (structuur) – néerlandais" lang="nl" hreflang="nl" data-title="Algebra (structuur)" data-language-autonym="Nederlands" data-language-local-name="néerlandais" 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/Algebra_over_ein_kropp" title="Algebra over ein kropp – norvégien nynorsk" lang="nn" hreflang="nn" data-title="Algebra over ein kropp" data-language-autonym="Norsk nynorsk" data-language-local-name="norvégien nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_nad_cia%C5%82em" title="Algebra nad ciałem – polonais" lang="pl" hreflang="pl" data-title="Algebra nad ciałem" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_sobre_um_corpo" title="Álgebra sobre um corpo – portugais" lang="pt" hreflang="pt" data-title="Álgebra sobre um corpo" data-language-autonym="Português" data-language-local-name="portugais" 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/Algebr%C4%83_peste_un_corp" title="Algebră peste un corp – roumain" lang="ro" hreflang="ro" data-title="Algebră peste un corp" data-language-autonym="Română" data-language-local-name="roumain" 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%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BD%D0%B0%D0%B4_%D0%BF%D0%BE%D0%BB%D0%B5%D0%BC" title="Алгебра над полем – russe" lang="ru" hreflang="ru" data-title="Алгебра над полем" data-language-autonym="Русский" data-language-local-name="russe" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Algebra_%C3%B6ver_en_kropp" title="Algebra över en kropp – suédois" lang="sv" hreflang="sv" data-title="Algebra över en kropp" data-language-autonym="Svenska" data-language-local-name="suédois" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BD%D0%B0%D0%B4_%D0%BF%D0%BE%D0%BB%D0%B5%D0%BC" title="Алгебра над полем – ukrainien" lang="uk" hreflang="uk" data-title="Алгебра над полем" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_tr%C3%AAn_m%E1%BB%99t_tr%C6%B0%E1%BB%9Dng" title="Đại số trên một trường – vietnamien" lang="vi" hreflang="vi" data-title="Đại số trên một trường" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamien" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9F%9F%E4%B8%8A%E7%9A%84%E4%BB%A3%E6%95%B0" title="域上的代数 – chinois" lang="zh" hreflang="zh" data-title="域上的代数" data-language-autonym="中文" data-language-local-name="chinois" 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/%E4%BB%A3%E6%95%B8_(%E4%BB%A3%E6%95%B8)" title="代數 (代數) – chinois littéraire" lang="lzh" hreflang="lzh" data-title="代數 (代數)" data-language-autonym="文言" data-language-local-name="chinois littéraire" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BB%A3%E6%95%B8_(%E4%BB%A3%E6%95%B8%E7%B5%90%E6%A7%8B)" title="代數 (代數結構) – cantonais" lang="yue" hreflang="yue" data-title="代數 (代數結構)" data-language-autonym="粵語" data-language-local-name="cantonais" 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/Q1000660#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> <div 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style="display:table-cell;padding-right:0.5em"> <p>Pour les articles homonymes, voir <a href="/wiki/Alg%C3%A8bre_(homonymie)" class="mw-disambig" title="Algèbre (homonymie)">Algèbre (homonymie)</a>. </p> </div></div> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, et plus précisément en <a href="/wiki/Alg%C3%A8bre_g%C3%A9n%C3%A9rale" title="Algèbre générale">algèbre générale</a>, une <b><a href="/wiki/Alg%C3%A8bre_sur_un_anneau" title="Algèbre sur un anneau">algèbre</a></b> sur un <a href="/wiki/Corps_commutatif" title="Corps commutatif">corps commutatif</a> <i>K</i>, ou simplement une <i>K</i>-<b>algèbre</b>, est une <a href="/wiki/Structure_alg%C3%A9brique" title="Structure algébrique">structure algébrique</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="{\textstyle (A,+,\cdot ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (A,+,\cdot ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3523f0ca2d8921455d02c12e4a9746dc2aa8c55b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.917ex; height:2.843ex;" alt="{\textstyle (A,+,\cdot ,\times )}"></span> telle que : </p> <ol><li>(<i>A</i>, +, ·) est un <a href="/wiki/Espace_vectoriel" title="Espace vectoriel">espace vectoriel</a> sur <i>K</i> ;</li> <li>la loi × est <a href="/wiki/Bilin%C3%A9aire" class="mw-redirect" title="Bilinéaire">K-bilinéaire</a>.</li></ol> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Définitions"><span id="D.C3.A9finitions"></span>Définitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=1" title="Modifier la section : Définitions" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=1" title="Modifier le code source de la section : Définitions"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une algèbre sur un corps commutatif <i>K</i> est un <i>K</i>-espace vectoriel <i>A</i> muni d'une <a href="/wiki/Op%C3%A9rateur_(math%C3%A9matiques)" title="Opérateur (mathématiques)">opération binaire</a> × (c'est-à-dire que le « produit » <span class="nowrap"><i>x</i> × <i>y</i></span> de deux éléments de <i>A</i> est un élément de <i>A</i>) bilinéaire, ce qui signifie que pour tous vecteurs <i>x</i>, <i>y</i>, <i>z</i> dans <i>A</i> et tous scalaires <i>a</i>, <i>b</i> dans <i>K</i>, les égalités suivantes sont vraies : </p> <ul><li>(<i>x + y</i>) × <i>z = x </i>× <i>z </i>+ <i>y </i>× <i>z</i> ;</li> <li><i>x </i>× (<i>y + z</i>) = <i>x </i>× <i>y </i>+ <i>x </i>× <i>z</i> ;</li> <li>(<i>a x</i>) × (<i>b y</i>) = (<i>a b</i>) (<i>x </i>× <i>y</i>).</li></ul> <p>Les deux premières égalités traduisent la <a href="/wiki/Distributivit%C3%A9" title="Distributivité">distributivité</a> de la loi × par rapport à la loi +. </p><p>On dit que <i>K</i> est le corps de base de <i>A</i>. L'opérateur binaire est souvent désigné comme la multiplication dans <i>A</i>. </p><p>Un morphisme entre deux algèbres <i>A</i> et <i>B</i> sur <i>K</i> est une <a href="/wiki/Application_(math%C3%A9matiques)" title="Application (mathématiques)">application</a> <span class="nowrap"><i>f </i>: <i>A</i> → <i>B</i></span> telle que <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x,y\in A,\,\forall a\in K,f(x\times y)=f(x)\times f(y){\textrm {et}}f(x+ay)=f(x)+af(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>∈</mo> <mi>K</mi> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>×</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>et</mtext> </mrow> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>a</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x,y\in A,\,\forall a\in K,f(x\times y)=f(x)\times f(y){\textrm {et}}f(x+ay)=f(x)+af(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58627ceaa775994072efd7156ac0cca530611834" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.349ex; height:2.843ex;" alt="{\displaystyle \forall x,y\in A,\,\forall a\in K,f(x\times y)=f(x)\times f(y){\textrm {et}}f(x+ay)=f(x)+af(y).}"></span> Deux algèbres <i>A</i> et <i>B</i> sur <i>K</i> sont dites <a href="/wiki/Isomorphisme" title="Isomorphisme">isomorphes</a> s'il existe une <a href="/wiki/Bijection" title="Bijection">bijection</a> de <i>A</i> dans <i>B</i> qui soit un morphisme d'algèbres. </p> <div class="mw-heading mw-heading2"><h2 id="Généralisation"><span id="G.C3.A9n.C3.A9ralisation"></span>Généralisation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=2" title="Modifier la section : Généralisation" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=2" title="Modifier le code source de la section : Généralisation"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans la définition, <i>K</i> peut être un <a href="/wiki/Anneau_commutatif" title="Anneau commutatif">anneau commutatif</a> unitaire, et <i>A</i> un <i>K</i>-<a href="/wiki/Module_sur_un_anneau" title="Module sur un anneau">module</a>. Alors, <i>A</i> est encore appelée une <i>K</i>-algèbre et on dit que <i>K</i> est l'anneau de base de <i>A</i>. </p> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Alg%C3%A8bre_sur_un_anneau" title="Algèbre sur un anneau">algèbre sur un anneau</a>.</div></div> <div class="mw-heading mw-heading2"><h2 id="Algèbres_associatives,_algèbres_commutatives_et_algèbres_unifères"><span id="Alg.C3.A8bres_associatives.2C_alg.C3.A8bres_commutatives_et_alg.C3.A8bres_unif.C3.A8res"></span>Algèbres associatives, algèbres commutatives et algèbres unifères</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=3" title="Modifier la section : Algèbres associatives, algèbres commutatives et algèbres unifères" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=3" title="Modifier le code source de la section : Algèbres associatives, algèbres commutatives et algèbres unifères"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Une <a href="/wiki/Alg%C3%A8bre_associative" title="Algèbre associative">algèbre associative</a> est une algèbre sur un anneau dont la loi de composition interne × est <a href="/wiki/Associativit%C3%A9" title="Associativité">associative</a>. Lorsque cet anneau est un corps, il s'agit donc d'une <a href="/wiki/Alg%C3%A8bre_associative_sur_un_corps" title="Algèbre associative sur un corps">algèbre associative sur un corps</a>.</li> <li>Une algèbre commutative est une algèbre sur un anneau dont la loi de composition interne × est <a href="/wiki/Commutativit%C3%A9" class="mw-redirect" title="Commutativité">commutative</a>.</li> <li>Une algèbre <a href="/wiki/Anneau_unif%C3%A8re#Note_terminologique_:_«_anneaux_»_sans_neutre_multiplicatif" class="mw-redirect" title="Anneau unifère">unifère</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup> est une algèbre sur un anneau dont la loi de composition interne × admet un <a href="/wiki/%C3%89l%C3%A9ment_neutre" title="Élément neutre">élément neutre</a>, noté <span class="texhtml">1</span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Bases_et_tables_de_multiplication_d'une_algèbre_sur_un_corps"><span id="Bases_et_tables_de_multiplication_d.27une_alg.C3.A8bre_sur_un_corps"></span>Bases et tables de multiplication d'une algèbre sur un corps</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=4" title="Modifier la section : Bases et tables de multiplication d'une algèbre sur un corps" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=4" title="Modifier le code source de la section : Bases et tables de multiplication d'une algèbre sur un corps"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Une <i>base</i> d'une algèbre <i>A</i> sur un corps <i>K</i> est une <a href="/wiki/Base_(alg%C3%A8bre_lin%C3%A9aire)" title="Base (algèbre linéaire)">base</a> de <i>A</i> pour sa structure d'espace vectoriel<sup id="cite_ref-bourbakiIIIp10_2-0" class="reference"><a href="#cite_note-bourbakiIIIp10-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p><p>Si <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=(a_{k})_{k\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(a_{k})_{k\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ac410f8ce136b324ab3ee7ad0dfddd43de7cfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.47ex; height:2.843ex;" alt="{\displaystyle a=(a_{k})_{k\in I}}"></span> est une base de <i>A</i>, il existe alors une unique famille <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 (c_{i,j}^{k})_{i,j,k\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{i,j}^{k})_{i,j,k\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e78f5ce3dd4e46b56ec0d325e8ad9d4d848b283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.924ex; height:3.509ex;" alt="{\displaystyle (c_{i,j}^{k})_{i,j,k\in I}}"></span> d'éléments du corps <i>K</i> tels que : <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\times a_{j}=\sum _{k\in I}c_{i,j}^{k}a_{k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\times a_{j}=\sum _{k\in I}c_{i,j}^{k}a_{k}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fee7e603d9d4e19003c8c7a605dcdfae13e75ee" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.757ex; height:5.676ex;" alt="{\displaystyle a_{i}\times a_{j}=\sum _{k\in I}c_{i,j}^{k}a_{k}.}"></span> </p><p>Pour <span class="texhtml"><i>i</i></span> et <span class="texhtml"><i>j</i></span> fixés, les coefficients sont nuls sauf un nombre fini d'entre eux. On dit 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 (c_{i,j}^{k})_{i,j,k\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (c_{i,j}^{k})_{i,j,k\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e78f5ce3dd4e46b56ec0d325e8ad9d4d848b283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.924ex; height:3.509ex;" alt="{\displaystyle (c_{i,j}^{k})_{i,j,k\in I}}"></span> sont les <b>constantes de structure</b><sup id="cite_ref-bourbakiIIIp10_2-1" class="reference"><a href="#cite_note-bourbakiIIIp10-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> de l'algèbre <i>A</i> par rapport à la base <span class="texhtml mvar" style="font-style:italic;">a</span>, et que les relations <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\times a_{j}=\sum _{k\in I}c_{i,j}^{k}a_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>×</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msubsup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\times a_{j}=\sum _{k\in I}c_{i,j}^{k}a_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16753d1ea6c2832a7d0354bdd71ed4ecca835552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.11ex; height:5.676ex;" alt="{\displaystyle a_{i}\times a_{j}=\sum _{k\in I}c_{i,j}^{k}a_{k}}"></span> constituent la <b>table de multiplication</b> de l'algèbre <i>A</i> pour la base <span class="texhtml mvar" style="font-style:italic;">a</span><sup id="cite_ref-bourbakiIIIp10_2-2" class="reference"><a href="#cite_note-bourbakiIIIp10-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup>. </p> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone"><span class="mw-valign-text-top noviewer" typeof="mw:File"><a href="/wiki/Fichier:Fairytale_warning.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Fairytale_warning.png/17px-Fairytale_warning.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Fairytale_warning.png/26px-Fairytale_warning.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Fairytale_warning.png/34px-Fairytale_warning.png 2x" data-file-width="64" data-file-height="64" /></a></span></div><div class="bandeau-cell">Cette section est vide, insuffisamment détaillée ou incomplète. <a href="/wiki/Sp%C3%A9cial:EditPage/Alg%C3%A8bre_sur_un_corps" title="Spécial:EditPage/Algèbre sur un corps">Votre aide</a> est la bienvenue ! <a href="/wiki/Aide:Comment_modifier_une_page" title="Aide:Comment modifier une page">Comment faire ?</a></div></div> <div class="mw-heading mw-heading2"><h2 id="Exemple_d'algèbre_de_dimension_infinie"><span id="Exemple_d.27alg.C3.A8bre_de_dimension_infinie"></span>Exemple d'algèbre de dimension infinie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=5" title="Modifier la section : Exemple d'algèbre de dimension infinie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=5" title="Modifier le code source de la section : Exemple d'algèbre de dimension infinie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> un <a href="/wiki/Ouvert_(topologie)" title="Ouvert (topologie)">ouvert</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 {\mathbb {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </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/ceb2c16f63cbe057c2cb7b44530e3b87865e8819" 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>. L'ensemble des <a href="/wiki/Analyse_complexe" title="Analyse complexe">fonctions analytiques</a> dans <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> est une <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </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/ceb2c16f63cbe057c2cb7b44530e3b87865e8819" 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>-algèbre. </p> <div class="mw-heading mw-heading2"><h2 id="Exemples_d'algèbres_de_dimension_finie"><span id="Exemples_d.27alg.C3.A8bres_de_dimension_finie"></span>Exemples d'algèbres de dimension finie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=6" title="Modifier la section : Exemples d'algèbres de dimension finie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=6" title="Modifier le code source de la section : Exemples d'algèbres de dimension finie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Algèbres_associatives_et_commutatives"><span id="Alg.C3.A8bres_associatives_et_commutatives"></span>Algèbres associatives et commutatives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=7" title="Modifier la section : Algèbres associatives et commutatives" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=7" title="Modifier le code source de la section : Algèbres associatives et commutatives"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Nombres_complexes">Nombres complexes</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=8" title="Modifier la section : Nombres complexes" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=8" title="Modifier le code source de la section : Nombres complexes"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'ensemble des <a href="/wiki/Nombres_complexes" class="mw-redirect" title="Nombres complexes">nombres complexes</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="{\textstyle (\mathbb {C} ,+,\cdot ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\mathbb {C} ,+,\cdot ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c7fcec6810bd945acd3b12e9da3a0b5474e6b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.852ex; height:2.843ex;" alt="{\textstyle (\mathbb {C} ,+,\cdot ,\times )}"></span> est une ℝ-algèbre <a href="/wiki/Associativit%C3%A9" title="Associativité">associative</a>, unifère et <a href="/wiki/Commutativit%C3%A9" class="mw-redirect" title="Commutativité">commutative</a> de dimension 2. Une base de l'algèbre ℂ est constituée des éléments 1 et i. La table de multiplication est constituée des relations : </p> <table class="wikitable centre"> <tbody><tr> <th width="40"></th> <th>1</th> <th>i </th></tr> <tr> <th>1 </th> <td>1 × 1 = 1 </td> <td>1 × i = i </td></tr> <tr> <th>i </th> <td>i × 1 = i </td> <td>i × i = –1 </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Corps_finis">Corps finis</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=9" title="Modifier la section : Corps finis" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=9" title="Modifier le code source de la section : Corps finis"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Tout <a href="/wiki/Corps_fini" title="Corps fini">corps fini</a> est une algèbre associative, unifère et commutative de dimension <i>n</i> sur son sous-corps premier (<b>F</b><sub><i>p</i></sub> = ℤ/<i>p</i>ℤ), donc son ordre est <i>p<sup>n</sup></i>. </p><p>Par exemple le corps fini <b>F</b><sub>4</sub> est une algèbre de dimension 2 sur le corps <b>F</b><sub>2</sub> = ℤ/2ℤ dont la table de multiplication dans une base (1, a) est : </p> <table class="wikitable centre"> <tbody><tr> <th width="40"></th> <th>1</th> <th>a </th></tr> <tr> <th>1 </th> <td>1 × 1 = 1 </td> <td>1 × a = a </td></tr> <tr> <th>a </th> <td>a × 1 = a </td> <td>a × a = 1 + a </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Algèbres_quadratiques"><span id="Alg.C3.A8bres_quadratiques"></span>Algèbres quadratiques</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=10" title="Modifier la section : Algèbres quadratiques" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=10" title="Modifier le code source de la section : Algèbres quadratiques"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On peut démontrer que toute algèbre unifère de dimension 2 sur un corps est associative et commutative<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. Sa table de multiplication dans une base (1, <i>x</i>) est de la forme : </p> <table class="wikitable centre"> <tbody><tr> <th width="40"></th> <th>1</th> <th><i>x</i> </th></tr> <tr> <th>1 </th> <td>1 × 1 = 1 </td> <td>1 × <i>x</i> = <i>x</i> </td></tr> <tr> <th><i>x</i> </th> <td><i>x</i> × 1 = <i>x</i> </td> <td><i>x</i> × <i>x</i> = <i>a</i>1 + <i>bx</i> </td></tr></tbody></table> <p>Une telle algèbre est appelée <b>algèbre quadratique</b> de type (<i>a</i>, <i>b</i>) (le type pouvant dépendre de la base choisie). </p><p>Par exemple : ℂ est une ℝ-algèbre quadratique de type (–1, 0) pour la base (1, i) et <b>F</b><sub>4</sub> est une <b>F</b><sub>2</sub>-algèbre quadratique de type (1, 1). </p> <div class="mw-heading mw-heading3"><h3 id="Algèbres_associatives_et_non_commutatives"><span id="Alg.C3.A8bres_associatives_et_non_commutatives"></span>Algèbres associatives et non commutatives</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=11" title="Modifier la section : Algèbres associatives et non commutatives" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=11" title="Modifier le code source de la section : Algèbres associatives et non commutatives"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Matrices_carrées"><span id="Matrices_carr.C3.A9es"></span>Matrices carrées</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=12" title="Modifier la section : Matrices carrées" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=12" title="Modifier le code source de la section : Matrices carrées"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Matrice_carr%C3%A9e" class="mw-redirect" title="Matrice carrée">Matrice carrée</a>.</div></div> <p>L'ensemble <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathcal {M}}_{n}(\mathbb {R} ),+,\cdot ,\times \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {M}}_{n}(\mathbb {R} ),+,\cdot ,\times \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/160c5e227832ce81c4f1e0d7043542fd7d24e32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.671ex; height:2.843ex;" alt="{\displaystyle \left({\mathcal {M}}_{n}(\mathbb {R} ),+,\cdot ,\times \right)}"></span> des <a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">matrices</a> carrées d'ordre <i>n </i>≥ 2 à coefficients réels est une ℝ-algèbre associative, unifère et <b>non</b> commutative de dimension <i>n</i><sup>2</sup>. </p> <div class="mw-heading mw-heading4"><h4 id="Quaternions">Quaternions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=13" title="Modifier la section : Quaternions" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=13" title="Modifier le code source de la section : Quaternions"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Quaternion" title="Quaternion">Quaternion</a>.</div></div> <p>L'ensemble <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\mathbb {H} ,+,\cdot ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\mathbb {H} ,+,\cdot ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fecb211069a395dde99fe08f540b97b00438717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.982ex; height:2.843ex;" alt="{\textstyle (\mathbb {H} ,+,\cdot ,\times )}"></span> des <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> est une ℝ-algèbre associative, unifère et <b>non</b> commutative de dimension 4. </p> <table class="wikitable centre" style="text-align:center;"> <tbody><tr> <th width="30"></th> <th>1</th> <th>i</th> <th>j</th> <th>k </th></tr> <tr> <th>1 </th> <td>1 × 1 = 1</td> <td>1 × i = i</td> <td>1 × j = j</td> <td>1 × k = k </td></tr> <tr> <th>i </th> <td>i × 1 = i</td> <td>i × i = –1</td> <td>i × j = k</td> <td>i × k = –j </td></tr> <tr> <th>j </th> <td>j × 1 = j</td> <td>j × i = –k</td> <td>j × j = –1</td> <td>j × k = i </td></tr> <tr> <th>k </th> <td>k × 1 = k</td> <td>k × i = j</td> <td>k × j = –i</td> <td>k × k = –1 </td></tr> </tbody></table> <div class="mw-heading mw-heading4"><h4 id="Biquaternions">Biquaternions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=14" title="Modifier la section : Biquaternions" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=14" title="Modifier le code source de la section : Biquaternions"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>L'ensemble <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\mathbb {B} ,+,\cdot ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">B</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\mathbb {B} ,+,\cdot ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdb5d42d6ee88d5fadad7fc82100f27d8a0d8b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.725ex; height:2.843ex;" alt="{\textstyle (\mathbb {B} ,+,\cdot ,\times )}"></span> des <a href="/wiki/Biquaternion" title="Biquaternion">biquaternions</a> est une ℂ-algèbre associative, unifère et <b>non</b> commutative de dimension 4 qui est isomorphe à l'algèbre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\mathcal {M}}_{2}(\mathbb {C} ),+,\cdot ,\times \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\mathcal {M}}_{2}(\mathbb {C} ),+,\cdot ,\times \right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6776f383a880a81bfa6c3e746492fc5f9b8fde2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.506ex; height:2.843ex;" alt="{\displaystyle \left({\mathcal {M}}_{2}(\mathbb {C} ),+,\cdot ,\times \right)}"></span> des <a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">matrices</a> carrées d'ordre 2 à coefficients complexes. </p> <div class="mw-heading mw-heading3"><h3 id="Algèbre_unifère_non_associative"><span id="Alg.C3.A8bre_unif.C3.A8re_non_associative"></span>Algèbre unifère non associative</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=15" title="Modifier la section : Algèbre unifère non associative" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=15" title="Modifier le code source de la section : Algèbre unifère non associative"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Octonion" title="Octonion">Octonion</a>.</div></div> <p>L'ensemble des <a href="/wiki/Octonion" title="Octonion">octonions</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\mathbb {O} ,+,\cdot ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">O</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\mathbb {O} ,+,\cdot ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b6eb39cd0fa2cc957baf654e84e7d6c3d8e947" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.982ex; height:2.843ex;" alt="{\textstyle (\mathbb {O} ,+,\cdot ,\times )}"></span> est une ℝ-algèbre unifère <b>non</b> associative et <b>non</b> commutative de dimension 8. </p> <div class="mw-heading mw-heading3"><h3 id="Algèbres_non_associatives_et_non_unifères"><span id="Alg.C3.A8bres_non_associatives_et_non_unif.C3.A8res"></span>Algèbres non associatives et non unifères</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=16" title="Modifier la section : Algèbres non associatives et non unifères" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=16" title="Modifier le code source de la section : Algèbres non associatives et non unifères"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Produit_vectoriel">Produit vectoriel</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=17" title="Modifier la section : Produit vectoriel" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=17" title="Modifier le code source de la section : Produit vectoriel"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Produit_vectoriel" title="Produit vectoriel">Produit vectoriel</a>.</div></div> <p>L'<a href="/wiki/Espace_euclidien" title="Espace euclidien">espace euclidien</a> ℝ<sup>3</sup> muni du <a href="/wiki/Produit_vectoriel" title="Produit vectoriel">produit vectoriel</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="{\textstyle (\mathbb {R} ^{3},+,\cdot ,\wedge )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>∧</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\mathbb {R} ^{3},+,\cdot ,\wedge )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5153b7d4d76671a4fd785557748024a8f9cd85c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.649ex; height:3.176ex;" alt="{\textstyle (\mathbb {R} ^{3},+,\cdot ,\wedge )}"></span>, est une ℝ-algèbre <b>non</b> associative, <b>non</b> unifère et <b>non</b> commutative (elle est anti-commutative) de dimension 3. </p><p>La table de multiplication dans une <a href="/wiki/Base_orthonormale" class="mw-redirect" title="Base orthonormale">base orthonormale</a> directe <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 ({\vec {u}},{\vec {v}},{\vec {w}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\vec {u}},{\vec {v}},{\vec {w}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc05226d8f4d938472aa2aa0a04c6d56ae20d8d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.046ex; height:2.843ex;" alt="{\displaystyle ({\vec {u}},{\vec {v}},{\vec {w}})}"></span> est : </p> <table class="wikitable centre"> <tbody><tr> <th width="50"></th> <th><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 {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span></th> <th><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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span></th> <th><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 {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}}"></span> </th></tr> <tr> <th><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 {\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89c41e9cf70c5e5b56e2128a136985a75f90ba43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}}"></span> </th> <td><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 {\vec {u}}\wedge {\vec {u}}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}\wedge {\vec {u}}={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7d9d1af23558fdc828ce9040afa7c8fc797dc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.503ex; height:2.843ex;" alt="{\displaystyle {\vec {u}}\wedge {\vec {u}}={\vec {0}}}"></span> </td> <td><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 {\vec {u}}\wedge {\vec {v}}={\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}\wedge {\vec {v}}={\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212022e32af2c44c3343b46803f37fb182d3e81f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.85ex; height:2.343ex;" alt="{\displaystyle {\vec {u}}\wedge {\vec {v}}={\vec {w}}}"></span> </td> <td><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 {\vec {u}}\wedge {\vec {w}}=-{\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {u}}\wedge {\vec {w}}=-{\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e80b1ca31b54f722219b0c1786ce7a9a1d3767a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle {\vec {u}}\wedge {\vec {w}}=-{\vec {v}}}"></span> </td></tr> <tr> <th><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 {\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85820588abd7333ef4d0c56539cb31c20e730753" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.175ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}}"></span> </th> <td><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 {\vec {v}}\wedge {\vec {u}}=-{\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\wedge {\vec {u}}=-{\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bfb1a9ce895aadbd97c3a223a7ecc44f41ed8e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle {\vec {v}}\wedge {\vec {u}}=-{\vec {w}}}"></span> </td> <td><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 {\vec {v}}\wedge {\vec {v}}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\wedge {\vec {v}}={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f039901f797a1cbed44247b9fa3453b5fc2b62ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.194ex; height:2.843ex;" alt="{\displaystyle {\vec {v}}\wedge {\vec {v}}={\vec {0}}}"></span> </td> <td><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 {\vec {v}}\wedge {\vec {w}}={\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {v}}\wedge {\vec {w}}={\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c22347ed654bd1386cac309cc6361d53f60be278" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.85ex; height:2.343ex;" alt="{\displaystyle {\vec {v}}\wedge {\vec {w}}={\vec {u}}}"></span> </td></tr> <tr> <th><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 {\vec {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b6c48cdaecf8d81481ea21b1d0c046bf34b68ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}}"></span> </th> <td><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 {\vec {w}}\wedge {\vec {u}}={\vec {v}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}\wedge {\vec {u}}={\vec {v}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e42ee51c51f1803a1858a582cc3a25aeea899ccf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.85ex; height:2.343ex;" alt="{\displaystyle {\vec {w}}\wedge {\vec {u}}={\vec {v}}}"></span> </td> <td><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 {\vec {w}}\wedge {\vec {v}}=-{\vec {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}\wedge {\vec {v}}=-{\vec {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed5765d08b2af31d3c7526f08ff3501f50c8fc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.658ex; height:2.509ex;" alt="{\displaystyle {\vec {w}}\wedge {\vec {v}}=-{\vec {u}}}"></span> </td> <td><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 {\vec {w}}\wedge {\vec {w}}={\vec {0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {w}}\wedge {\vec {w}}={\vec {0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/364357ec23b902a01797d13eeb80b98dc26fd158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.172ex; height:2.843ex;" alt="{\displaystyle {\vec {w}}\wedge {\vec {w}}={\vec {0}}}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading4"><h4 id="Crochet_de_Lie">Crochet de Lie</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=18" title="Modifier la section : Crochet de Lie" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=18" title="Modifier le code source de la section : Crochet de Lie"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="bandeau-container bandeau-section metadata bandeau-niveau-information"><div class="bandeau-cell bandeau-icone-css loupe">Article détaillé : <a href="/wiki/Crochet_de_Lie" title="Crochet de Lie">Crochet de Lie</a>.</div></div> <p>L'ensemble des <a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">matrices</a> carrées d'ordre <i>n </i>≥ 2 à coefficients réels, muni du <a href="/wiki/Crochet_de_Lie" title="Crochet de Lie">crochet de Lie</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 [M,N]=MN-NM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>M</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>M</mi> <mi>N</mi> <mo>−</mo> <mi>N</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [M,N]=MN-NM}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9de391cf54d93f3384d0cc9af5427685ff879ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.784ex; height:2.843ex;" alt="{\displaystyle [M,N]=MN-NM}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left({\mathcal {M}}_{n}(\mathbb {R} ),+,\cdot ,[,]\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo stretchy="false">[</mo> <mo>,</mo> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left({\mathcal {M}}_{n}(\mathbb {R} ),+,\cdot ,[,]\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bfa42cc9c9b7072433c9cfc6948b1159e54df48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.19ex; height:2.843ex;" alt="{\textstyle \left({\mathcal {M}}_{n}(\mathbb {R} ),+,\cdot ,[,]\right)}"></span> est une ℝ-algèbre <b>non</b> associative, <b>non</b> unifère et <b>non</b> commutative de dimension <i>n</i><sup>2</sup>. Elle est anti-commutative et possède des propriétés qui font de l'algèbre une <a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">algèbre de Lie</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Contre-exemple">Contre-exemple</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=19" title="Modifier la section : Contre-exemple" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=19" title="Modifier le code source de la section : Contre-exemple"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La ℝ-algèbre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\mathbb {H} ,+,\cdot ,\times )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo>,</mo> <mo>×</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle (\mathbb {H} ,+,\cdot ,\times )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fecb211069a395dde99fe08f540b97b00438717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.982ex; height:2.843ex;" alt="{\textstyle (\mathbb {H} ,+,\cdot ,\times )}"></span> des <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> est un ℂ-espace vectoriel, mais n'est pas une ℂ-algèbre car la multiplication × n'est pas ℂ-bilinéaire : <span class="texhtml">i·(j × k) ≠ j × (i·k)</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=20" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=20" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r194021218">.mw-parser-output .autres-projets>.titre{text-align:center;margin:0.2em 0}.mw-parser-output .autres-projets>ul{margin:0;padding:0}.mw-parser-output .autres-projets>ul>li{list-style:none;margin:0.2em 0;text-indent:0;padding-left:24px;min-height:20px;text-align:left;display:block}.mw-parser-output .autres-projets>ul>li>a{font-style:italic}@media(max-width:720px){.mw-parser-output .autres-projets{float:none}}</style><div class="autres-projets boite-grise boite-a-droite noprint js-interprojets"> <p class="titre">Sur les autres projets Wikimedia :</p> <ul class="noarchive plainlinks"> <li class="wikiversity"><a href="https://fr.wikiversity.org/wiki/Alg%C3%A8bre_sur_un_corps" class="extiw" title="v:Algèbre sur un corps">Algèbre sur un corps</a>, <span class="nowrap">sur <span class="project">Wikiversity</span></span></li> </ul> </div> <ul><li><a href="/wiki/Alg%C3%A8bre_de_Clifford" title="Algèbre de Clifford">Algèbre de Clifford</a></li> <li><a href="/wiki/Alg%C3%A8bre_g%C3%A9om%C3%A9trique_(structure)" title="Algèbre géométrique (structure)">Algèbre géométrique</a></li> <li><a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">Algèbre de Lie</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&veaction=edit&section=21" title="Modifier la section : Notes et références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Alg%C3%A8bre_sur_un_corps&action=edit&section=21" title="Modifier le code source de la section : Notes et références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Bourbaki"><span class="ouvrage" id="N._Bourbaki"><a href="/wiki/N._Bourbaki" class="mw-redirect" title="N. Bourbaki">N. Bourbaki</a>, <cite class="italique"><a href="/wiki/%C3%89l%C3%A9ments_de_math%C3%A9matique" title="Éléments de mathématique">Théories spectrales</a></cite> <small style="line-height:1em;">(<a rel="nofollow" class="external text" href="//books.google.com/books?id=6AP4DrV5B4cC&pg=PA1">lire en ligne</a>)</small>, <abbr class="abbr" title="chapitre(s)">chap.</abbr> 1, <abbr class="abbr" title="page">p.</abbr> 1<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9ories+spectrales&rft.aulast=Bourbaki&rft.aufirst=N.&rft.pages=1&rft_id=%2F%2Fbooks.google.com%2Fbooks%3Fid%3D6AP4DrV5B4cC%26pg%3DPA1&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AAlg%C3%A8bre+sur+un+corps"></span></span></span>.</span> </li> <li id="cite_note-bourbakiIIIp10-2"><span class="mw-cite-backlink noprint">↑ <sup><a href="#cite_ref-bourbakiIIIp10_2-0">a</a> <a href="#cite_ref-bourbakiIIIp10_2-1">b</a> et <a href="#cite_ref-bourbakiIIIp10_2-2">c</a></sup> </span><span class="reference-text">N. Bourbaki, <i>Algèbre</i>, chapitre III, p. 10.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text">N. Bourbaki, <i>Algèbre</i>, chapitre III, p. 13, proposition 1.</span> </li> </ol></div> </div> <div class="navbox-container" style="clear:both;"> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="3" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Alg%C3%A8bre_lin%C3%A9aire" title="Modèle:Palette Algèbre linéaire"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Alg%C3%A8bre_lin%C3%A9aire&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Alg%C3%A8bre_lin%C3%A9aire" title="Algèbre linéaire">Algèbre linéaire</a> générale</div></th> </tr> <tr> <td class="navbox-banner" style="" colspan="3"><div class="liste-horizontale"> <ul><li><a href="/wiki/Vecteur" title="Vecteur">Vecteur</a></li> <li><a href="/wiki/Scalaire_(math%C3%A9matiques)" title="Scalaire (mathématiques)">Scalaire</a></li> <li><a href="/wiki/Combinaison_lin%C3%A9aire" title="Combinaison linéaire">Combinaison linéaire</a></li> <li><a href="/wiki/Espace_vectoriel" title="Espace vectoriel">Espace vectoriel</a></li> <li><a href="/wiki/Matrice_(math%C3%A9matiques)" title="Matrice (mathématiques)">Matrice</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Famille de vecteurs</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Famille_g%C3%A9n%C3%A9ratrice" title="Famille génératrice">Famille génératrice</a></li> <li><a href="/wiki/Ind%C3%A9pendance_lin%C3%A9aire" title="Indépendance linéaire">Famille libre (indépendance linéaire)</a></li> <li><a href="/wiki/Base_(alg%C3%A8bre_lin%C3%A9aire)" title="Base (algèbre linéaire)">Base</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_de_la_base_incompl%C3%A8te" title="Théorème de la base incomplète">Théorème de la base incomplète</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_de_la_dimension_pour_les_espaces_vectoriels" title="Théorème de la dimension pour les espaces vectoriels">Théorème de la dimension pour les espaces vectoriels</a></li> <li><a href="/wiki/Rang_(alg%C3%A8bre_lin%C3%A9aire)" title="Rang (algèbre linéaire)">Rang</a></li> <li><a href="/wiki/Colin%C3%A9arit%C3%A9" title="Colinéarité">Colinéarité</a></li></ul> </div></td> <td class="navbox-image" rowspan="6" style="vertical-align:middle;padding-left:7px"><span typeof="mw:File"><a href="/wiki/Fichier:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description" title="Mathématiques"><img alt="Mathématiques" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/80px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="80" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/120px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/160px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></td> </tr> <tr> <th class="navbox-group" style="">Sous-espace</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Sous-espace_vectoriel" title="Sous-espace vectoriel">Sous-espace vectoriel</a></li> <li><a href="/wiki/Somme_de_Minkowski" title="Somme de Minkowski">Somme de Minkowski</a></li> <li><a href="/wiki/Somme_directe" title="Somme directe">Somme directe</a></li> <li><a href="/wiki/Sous-espace_suppl%C3%A9mentaire" title="Sous-espace supplémentaire">Sous-espace supplémentaire</a></li> <li><a href="/wiki/Dimension_d%27un_espace_vectoriel" title="Dimension d'un espace vectoriel">Dimension</a></li> <li><a href="/wiki/Codimension" title="Codimension">Codimension</a></li> <li><a href="/wiki/Droite_vectorielle" title="Droite vectorielle">Droite</a></li> <li><a href="/wiki/Plan_vectoriel" title="Plan vectoriel">Plan</a></li> <li><a href="/wiki/Hyperplan" title="Hyperplan">Hyperplan</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Morphisme et<br />notions relatives</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Application_lin%C3%A9aire" title="Application linéaire">Application linéaire</a></li> <li><a href="/wiki/Noyau_(alg%C3%A8bre)" title="Noyau (algèbre)">Noyau</a></li> <li><a href="/wiki/Conoyau" title="Conoyau">Conoyau</a></li> <li><a href="/wiki/Lemme_des_noyaux" title="Lemme des noyaux">Lemme des noyaux</a></li> <li><a href="/wiki/Pseudo-inverse" title="Pseudo-inverse">Pseudo-inverse</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_de_factorisation" title="Théorème de factorisation">Théorème de factorisation</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_du_rang" title="Théorème du rang">Théorème du rang</a></li> <li><a href="/wiki/%C3%89quation_lin%C3%A9aire" title="Équation linéaire">Équation linéaire</a></li> <li><a href="/wiki/Syst%C3%A8me_d%27%C3%A9quations_lin%C3%A9aires" title="Système d'équations linéaires">Système d'équations linéaires</a></li> <li><a href="/wiki/%C3%89limination_de_Gauss-Jordan" title="Élimination de Gauss-Jordan">Élimination de Gauss-Jordan</a></li> <li><a href="/wiki/Forme_lin%C3%A9aire" title="Forme linéaire">Forme linéaire</a></li> <li><a href="/wiki/Espace_dual" title="Espace dual">Espace dual</a></li> <li><a href="/wiki/Orthogonalit%C3%A9" title="Orthogonalité">Orthogonalité</a></li> <li><a href="/wiki/Base_duale" title="Base duale">Base duale</a></li> <li><a href="/wiki/Endomorphisme_lin%C3%A9aire" title="Endomorphisme linéaire">Endomorphisme linéaire</a></li> <li><a href="/wiki/Valeur_propre,_vecteur_propre_et_espace_propre" title="Valeur propre, vecteur propre et espace propre">Valeur propre, vecteur propre et espace propre</a></li> <li><a href="/wiki/Projecteur_(math%C3%A9matiques)" title="Projecteur (mathématiques)">Projecteur</a></li> <li><a href="/wiki/Sym%C3%A9trie_vectorielle" title="Symétrie vectorielle">Symétrie</a></li> <li><a href="/wiki/Matrice_diagonalisable" title="Matrice diagonalisable">Matrice diagonalisable</a></li> <li><a href="/wiki/Diagonalisation" title="Diagonalisation">Diagonalisation</a></li> <li><a href="/wiki/Endomorphisme_nilpotent" title="Endomorphisme nilpotent">Endomorphisme nilpotent</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Dimension finie</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Espace_vectoriel_de_dimension_finie" title="Espace vectoriel de dimension finie">Espace vectoriel de dimension finie</a></li> <li><a href="/wiki/Trace_(alg%C3%A8bre)" title="Trace (algèbre)">Trace</a></li> <li><a href="/wiki/D%C3%A9terminant_(math%C3%A9matiques)" title="Déterminant (mathématiques)">Déterminant</a></li> <li><a href="/wiki/Polyn%C3%B4me_caract%C3%A9ristique" title="Polynôme caractéristique">Polynôme caractéristique</a></li> <li><a href="/wiki/Polyn%C3%B4me_d%27endomorphisme" title="Polynôme d'endomorphisme">Polynôme d'endomorphisme</a></li> <li><a href="/wiki/Th%C3%A9or%C3%A8me_de_Cayley-Hamilton" title="Théorème de Cayley-Hamilton">Théorème de Cayley-Hamilton</a></li> <li><a href="/wiki/Polyn%C3%B4me_minimal_d%27un_endomorphisme" title="Polynôme minimal d'un endomorphisme">Polynôme minimal d'un endomorphisme</a></li> <li><a href="/wiki/Invariants_de_similitude" title="Invariants de similitude">Invariants de similitude</a></li> <li><a href="/wiki/R%C3%A9duction_d%27endomorphisme" title="Réduction d'endomorphisme">Réduction d'endomorphisme</a></li> <li><a href="/wiki/R%C3%A9duction_de_Jordan" title="Réduction de Jordan">Réduction de Jordan</a></li> <li><a href="/wiki/D%C3%A9composition_de_Dunford" title="Décomposition de Dunford">Décomposition de Dunford</a></li> <li><a href="/wiki/D%C3%A9composition_de_Frobenius" title="Décomposition de Frobenius">Décomposition de Frobenius</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Enrichissements<br />de structure</th> <td class="navbox-list" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Norme_(math%C3%A9matiques)" title="Norme (mathématiques)">Norme</a></li> <li><a href="/wiki/Produit_scalaire" title="Produit scalaire">Produit scalaire</a></li> <li><a href="/wiki/Forme_quadratique" title="Forme quadratique">Forme quadratique</a></li> <li><a href="/wiki/Espace_vectoriel_topologique" title="Espace vectoriel topologique">Espace vectoriel topologique</a></li> <li><a href="/wiki/Orientation_(math%C3%A9matiques)" title="Orientation (mathématiques)">Orientation</a></li> <li><a class="mw-selflink selflink">Algèbre sur un corps</a></li> <li><a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">Algèbre de Lie</a></li> <li><a href="/wiki/Complexe_diff%C3%A9rentiel" title="Complexe différentiel">Complexe différentiel</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="">Développements</th> <td class="navbox-list navbox-even" style=""><div class="liste-horizontale"> <ul><li><a href="/wiki/Th%C3%A9orie_des_matrices" title="Théorie des matrices">Théorie des matrices</a></li> <li><a href="/wiki/Repr%C3%A9sentation_de_groupe" title="Représentation de groupe">Représentation de groupe</a></li> <li><a href="/wiki/Analyse_fonctionnelle_(math%C3%A9matiques)" title="Analyse fonctionnelle (mathématiques)">Analyse fonctionnelle</a></li> <li><a href="/wiki/Alg%C3%A8bre_multilin%C3%A9aire" title="Algèbre multilinéaire">Algèbre multilinéaire</a></li> <li><a href="/wiki/Module_sur_un_anneau" title="Module sur un anneau">Module sur un anneau</a></li></ul> </div></td> </tr> </tbody></table> <table class="navbox collapsible noprint autocollapse" style=""> <tbody><tr><th class="navbox-title" colspan="2" style=""><div style="float:left; width:6em; text-align:left"><div class="noprint plainlinks nowrap tnavbar" style="padding:0; font-size:xx-small; color:var(--color-emphasized, #000000);"><a href="/wiki/Mod%C3%A8le:Palette_Structures_alg%C3%A9briques" title="Modèle:Palette Structures algébriques"><abbr class="abbr" title="Voir ce modèle.">v</abbr></a> · <a class="external text" href="https://fr.wikipedia.org/w/index.php?title=Mod%C3%A8le:Palette_Structures_alg%C3%A9briques&action=edit"><abbr class="abbr" title="Modifier ce modèle. Merci de prévisualiser avant de sauvegarder.">m</abbr></a></div></div><div style="font-size:110%"><a href="/wiki/Structure_alg%C3%A9brique" title="Structure algébrique">Structures algébriques</a></div></th> </tr> <tr> <th class="navbox-group" style="width:40px">Pures</th> <td class="navbox-list" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="width:70px;"><a href="/wiki/Magma_(alg%C3%A8bre)" title="Magma (algèbre)">Magmas</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">Groupe</a></li> <li><a href="/wiki/Quasigroupe" title="Quasigroupe">Quasigroupe</a></li> <li><a href="/wiki/Demi-groupe" title="Demi-groupe">Demi-groupe</a></li> <li><a href="/wiki/Mono%C3%AFde" title="Monoïde">Monoïde</a></li> <li><a href="/wiki/Groupe_ab%C3%A9lien" title="Groupe abélien">Groupe abélien</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:70px;">Moduloïdes</th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Espace_vectoriel" title="Espace vectoriel">Espace vectoriel</a></li> <li><a href="/wiki/Espace_affine" title="Espace affine">Espace affine</a></li> <li><a href="/wiki/Groupe_%C3%A0_op%C3%A9rateurs" title="Groupe à opérateurs">Groupe à opérateurs</a></li> <li><a href="/wiki/Module_sur_un_anneau" title="Module sur un anneau">Module sur un anneau</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:70px;">Annélides</th> <td class="navbox-list navbox-even" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/w/index.php?title=Anneau_non_associatif&action=edit&redlink=1" class="new" title="Anneau non associatif (page inexistante)">Anneau non associatif</a> <a href="https://en.wikipedia.org/wiki/Nonassociative_ring" class="extiw" title="en:Nonassociative ring"><span class="indicateur-langue" title="Article en anglais : « Nonassociative ring »">(en)</span></a></li> <li><a href="/wiki/Pseudo-anneau" title="Pseudo-anneau">Pseudo-anneau</a></li> <li><a href="/wiki/Demi-anneau" title="Demi-anneau">Demi-anneau</a></li> <li><a href="/wiki/Dio%C3%AFde" title="Dioïde">Dioïde</a></li> <li><a href="/wiki/Anneau_(math%C3%A9matiques)" title="Anneau (mathématiques)">Anneau</a> <ul><li><a href="/wiki/Anneau_unitaire" title="Anneau unitaire">unitaire</a></li> <li><a href="/wiki/Anneau_commutatif" title="Anneau commutatif">commutatif</a></li> <li><a href="/wiki/Anneau_sans_diviseur_de_z%C3%A9ro" title="Anneau sans diviseur de zéro">sans diviseur de zéro</a></li> <li><a href="/wiki/Anneau_int%C3%A8gre" title="Anneau intègre">intègre</a></li></ul></li> <li><a href="/wiki/Corps_(math%C3%A9matiques)" title="Corps (mathématiques)">Corps</a> <ul><li><a href="/wiki/Corps_commutatif" title="Corps commutatif">commutatif</a></li> <li><a href="/wiki/Corps_gauche" title="Corps gauche">gauche</a></li></ul></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:70px;"><a href="/wiki/Alg%C3%A8bre_sur_un_anneau" title="Algèbre sur un anneau">Algèbre</a></th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Alg%C3%A8bre_associative" title="Algèbre associative">Algèbre associative</a></li> <li><a class="mw-selflink selflink">Algèbre sur un corps</a></li> <li><a href="/wiki/Alg%C3%A8bre_associative_sur_un_corps" title="Algèbre associative sur un corps">Algèbre associative sur un corps</a></li> <li><a href="/wiki/Alg%C3%A8bre_unitaire" title="Algèbre unitaire">Algèbre unitaire</a></li> <li><a href="/wiki/Alg%C3%A8bre_%C3%A0_division" title="Algèbre à division">Algèbre à division</a></li> <li><a href="/wiki/Alg%C3%A8bre_de_Clifford" title="Algèbre de Clifford">Algèbre de Clifford</a></li> <li><a href="/wiki/Alg%C3%A8bre_de_Jordan" title="Algèbre de Jordan">Algèbre de Jordan</a></li> <li><a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">Algèbre de Lie</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:70px;">Autres</th> <td class="navbox-list navbox-even" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Alg%C3%A8bre_de_Hopf" title="Algèbre de Hopf">Algèbre de Hopf</a></li> <li><a href="/wiki/Espace_homog%C3%A8ne" title="Espace homogène">Espace homogène</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> <tr> <th class="navbox-group" style="width:40px">Enrichies</th> <td class="navbox-list navbox-even" style=""><table class="navbox-subgroup" style=""> <tbody><tr> <th class="navbox-group" style="width:12em;">Espace topologique</th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Semi-groupe_topologique" title="Semi-groupe topologique">Semi-groupe topologique</a></li> <li><a href="/w/index.php?title=Mono%C3%AFde_topologique&action=edit&redlink=1" class="new" title="Monoïde topologique (page inexistante)">Monoïde topologique</a> <a href="https://en.wikipedia.org/wiki/Topological_monoid" class="extiw" title="en:Topological monoid"><span class="indicateur-langue" title="Article en anglais : « Topological monoid »">(en)</span></a></li> <li><a href="/wiki/Groupe_topologique" title="Groupe topologique">Groupe topologique</a></li> <li><a href="/wiki/Anneau_topologique" title="Anneau topologique">Anneau topologique</a></li> <li><a href="/wiki/Anneau_topologique" title="Anneau topologique">Corps topologique</a></li> <li><a href="/wiki/Corps_valu%C3%A9" title="Corps valué">Corps valué</a></li> <li><a href="/w/index.php?title=Module_topologique&action=edit&redlink=1" class="new" title="Module topologique (page inexistante)">Module topologique</a> <a href="https://en.wikipedia.org/wiki/Topological_module" class="extiw" title="en:Topological module"><span class="indicateur-langue" title="Article en anglais : « Topological module »">(en)</span></a></li> <li><a href="/wiki/Espace_vectoriel_topologique" title="Espace vectoriel topologique">Espace vectoriel topologique</a></li> <li><a href="/w/index.php?title=Alg%C3%A8bre_topologique&action=edit&redlink=1" class="new" title="Algèbre topologique (page inexistante)">Algèbre topologique</a> <a href="https://en.wikipedia.org/wiki/Topological_algebra" class="extiw" title="en:Topological algebra"><span class="indicateur-langue" title="Article en anglais : « Topological algebra »">(en)</span></a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:12em;"><a href="/wiki/Espace_m%C3%A9trique" title="Espace métrique">Espaces métriques</a></th> <td class="navbox-list navbox-even" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Espace_vectoriel_norm%C3%A9" title="Espace vectoriel normé">Espace vectoriel normé</a></li> <li><a href="/wiki/Espace_de_Banach" title="Espace de Banach">Espace de Banach</a></li> <li><a href="/wiki/Espace_pr%C3%A9hilbertien" title="Espace préhilbertien">Espace préhilbertien</a></li> <li><a href="/wiki/Espace_euclidien" title="Espace euclidien">Espace euclidien</a></li> <li><a href="/wiki/Espace_hermitien" title="Espace hermitien">Espace hermitien</a></li> <li><a href="/wiki/Espace_de_Hilbert" title="Espace de Hilbert">Espace de Hilbert</a></li></ul> </div></td> </tr> <tr> <th class="navbox-group" style="width:12em;">Géométrie différentielle et algébrique</th> <td class="navbox-list" style="text-align:left;;"><div class="liste-horizontale"> <ul><li><a href="/wiki/Groupe_de_Lie" title="Groupe de Lie">Groupe de Lie</a></li> <li><a href="/wiki/Groupe_alg%C3%A9brique" title="Groupe algébrique">Groupe algébrique</a></li></ul> </div></td> </tr> </tbody></table></td> </tr> </tbody></table> </div> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail de l’algèbre"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/24px-Arithmetic_symbols.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/36px-Arithmetic_symbols.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/48px-Arithmetic_symbols.svg.png 2x" data-file-width="210" data-file-height="210" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail:Algèbre">Portail de l’algèbre</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: 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