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Bibliographie </a> </li> </ul> <button class="btn btn-sm btn-universalis" data-bs-toggle="collapse" data-bs-target="#free-toc" aria-expanded="false" title="Afficher / masquer le sommaire"> <i aria-hidden="true" class="fa fa-fw fa-eye"></i> Sommaire </button> </div> <h1 class="mb-1"> <strong>ALG脠BRE</strong> </h1> <ul class="pagination justify-content-center public-links "><li class="page-item active"><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 1" class="page-link" title="Abonnez-vous" data-loaded="true">1</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 2" class="page-link" title="Abonnez-vous" data-loaded="true">2</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 3" class="page-link" title="Abonnez-vous" data-loaded="true">3</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 4" class="page-link" title="Abonnez-vous" data-loaded="true">4</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 5" class="page-link" title="Abonnez-vous" data-loaded="true">5</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 6" class="page-link" title="Abonnez-vous" data-loaded="true">6</a></li></ul><p class="text-sm text-muted mb-1">Article modifi茅 le <time datetime="2025-01-29"> 29/01/2025 </time></p><div class="content paged type-article" ><div class="block_1" data-ucode="yA911901_0"><i class="anchor" id="yA911901_0"></i><p> L' alg猫bre au sens moderne, 脿 savoir l'茅tude des structures alg茅briques ind茅pendamment de leurs r茅alisations concr猫tes, ne s'est d茅gag茅e que tr猫s progressivement au cours du <span class="pc">xix</span><sup>e</sup>聽si猫cle, en liaison avec le mouvement g茅n茅ral d'axiomatisation de l'ensemble des math茅matiques et la pr茅occupation croissante des math茅maticiens de 芦聽substituer les id茅es au calcul聽禄聽; jusqu'alors, le propos essentiel de l'alg猫bre avait 茅t茅 la r茅solution, par des formules explicites, des <a data-bs-toggle="popover" data-bs-content="D猫s la plus haute antiquit茅, on rencontre, 脿 l'occasion de probl猫mes concrets, des exemples de r茅solution d'茅quations du premier et du second degr茅, et, jusqu'au d茅but du xix e 聽si猫cle, l'茅tude des 茅quations constitue l'unique pr茅occupation des alg茅bristes.. Le d茅veloppement de la th茅orie est 茅troitement li茅 aux extensions successives de la notion de nombre 聽: introduction des..." title="Article : 脡QUATIONS ALG脡BRIQUES" href="/encyclopedie/equations-algebriques/">茅quations alg茅briques</a>. Les tentatives infructueuses pour r茅soudre les 茅quations g茅n茅rales de degr茅 sup茅rieur ou 茅gal 脿 cinq, ainsi que les probl猫mes de la th茅orie des nombres, conduisirent alors les math茅maticiens 脿 introduire des 锚tres math茅matiques de nature nouvelle qui pr茅sentaient entre eux des analogies 茅troites dans leur maniement et par suite 脿 ressentir le besoin de d茅gager ce qui pouvait 锚tre commun 脿 toutes ces situations. Ils furent ainsi amen茅s 脿 penser que la 芦聽nature聽禄 des objets math茅matiques 茅tudi茅s est au fond secondaire, et le math茅maticien anglais George Boole pouvait d茅clarer en 1847聽: 芦聽La <a data-bs-toggle="popover" data-bs-content="La math茅matique est une science hypoth茅tico-d茅ductive qui, en d茅veloppant un langage autonome, 茅labore et 茅tudie des notions abstraites li茅es les unes aux autres et souvent capables de fournir des mod猫les et des processus op茅ratoires permettant de mieux comprendre de nombreux aspects du monde observable, en particulier lorsque peuvent 锚tre invoqu茅es des id茅es de quantit茅, de forme et ..." title="Article : MATH脡MATIQUE" href="/encyclopedie/mathematique/">math茅matique</a> traite les op茅rations consid茅r茅es en elles-m锚mes, ind茅pendamment des mati猫res diverses auxquelles elles peuvent 锚tre appliqu茅es.聽禄</p><p>Tout au long du <span class="pc">xix</span><sup>e</sup>聽si猫cle va se d茅velopper ce processus d'axiomatisation de l'alg猫bre qui aboutit aux structures actuelles. Si, d猫s 1850, les math茅maticiens anglais ont d茅gag茅 avec une parfaite nettet茅 la notion de loi de composition et l'appliquent 脿 des situations vari茅es (vecteurs, matrices, alg猫bre de la logique), il faudra attendre 1910 pour trouver dans la vaste synth猫se de Steinitz l'expos茅 abstrait qui marque le d茅but de l'alg猫bre moderne proprement dite.</p><p>L'茅tude des groupes domine tout d'abord les pr茅occupations de cette 茅poque聽; introduite par Cauchy et surtout mise en 茅vidence par Galois qui en a montr茅 l'importance dans la th茅orie des 茅quations, cette notion va jouer un r么le essentiel dans presque tous les domaines des math茅matiques, en physique et en m茅canique quantique. Les travaux des math茅maticiens allemands sur les nombres alg茅briques seront 脿 l'origine de l'茅tude des corps et des <a data-bs-toggle="popover" data-bs-content="Dans tout ce qui suit, on se bornera 脿 consid茅rer des anneaux commutatifs unitaires, c'est-脿-dire poss茅dant un 茅l茅ment unit茅 pour la multiplication, not茅聽1. Les d茅finitions sont celles de l'article suivant, anneaux et alg猫bres .. De nombreux cas particuliers d'anneaux commutatifs unitaires ont 茅t茅茅tudi茅s au xix e 聽si猫cle, principalement 脿 propos de recherches de th茅orie des n..." title="Article : ANNEAUX COMMUTATIFS" href="/encyclopedie/anneaux-commutatifs/">anneaux commutatifs</a> et ces notions appara卯tront comme les outils essentiels pour 茅tudier les courbes et surfaces alg茅briques, conduisant 脿 la <a data-bs-toggle="popover" data-bs-content="La g茅om茅trie est commun茅ment d茅finie comme la science des figures de l'espace. Cette d茅finition un peu incertaine risque de conduire 脿 inclure dans la g茅om茅trie des questions qui ne sont g茅om茅triques que dans leur langage, mais rel猫vent en fait d'autres domaines. Tel est le cas de l' alg猫bre g茅om茅trique des Grecs qui parlait du 芦聽rectangle聽禄 de deux segments pour qualifier le pr..." title="Article : G脡OM脡TRIE" href="/encyclopedie/geometrie/">g茅om茅trie</a> alg茅brique abstraite聽; ainsi s'introduit le langage g茅om茅trique en alg猫bre commutative. L'<a data-bs-toggle="popover" data-bs-content="L' alg猫bre lin茅aire sur un corps commutatif, telle qu'on la trouvera pr茅sent茅e ici, s'est progressivement d茅gag茅e, au cours du xix e 聽si猫cle et au d茅but du xx e , de la th茅orie des 茅quations lin茅aires (syst猫mes de n 茅quations lin茅aires 脿 p inconnues, 茅quations diff茅rentielles et int茅grales lin茅aires) et de la g茅om茅trie (calcul vectoriel dans les espaces affines, ..." title="Article : ALG脠BRE LIN脡AIRE" href="/encyclopedie/lineaire-algebre/">alg猫bre lin茅aire</a> prend une grande importance lorsqu'apr猫s une axiomatisation convenable les math茅maticiens s'aper莽oivent du caract猫re lin茅aire de nombreuses situations et de l'importance du processus de lin茅arisation. Et comme 芦聽la math茅matique est un organisme dont la force vitale a pour condition l'indissoluble union de ses parties聽禄 (Hilbert, <em>Conclusion de la conf茅rence de 1900</em>), l'alg猫bre a rejoint avec succ猫s l'analyse par la consid茅ration simultan茅e, sur un m锚me ensemble, de structures alg茅briques et topologiques (constituant ainsi la branche des math茅matiques appel茅e alg猫bre topologique).</p><div class="clearfix"></div></div><div class="block_1" data-ucode="yA911901_1"><i class="anchor" id="yA911901_1"></i><h2>La th茅orie des groupes</h2><div class="block_2" data-ucode="yA911901_1_0"><i class="anchor" id="yA911901_1_0"></i><h3>La structure de groupe</h3><p> La structure de groupe est une des structures alg茅briques les plus simples et, sans conteste, la plus importante des math茅matiques modernes. Son universalit茅 ne s'arr锚te pas l脿聽: le psychologue Piaget a mis en 茅vidence le聽r么le essentiel jou茅 par cette notion dans les m茅canismes m锚mes de la pens茅e, et H.聽Poincar茅 a pu dire que la notion de groupe pr茅existe dans notre esprit car la g茅om茅trie ne se concevrait pas sans elle. Cependant, il a fallu presque un si猫cle pour que se d茅gage sous forme abstraite cette notion qui est maintenant introduite couramment dans l'enseignement secondaire.</p><p>Axiomatiquement, un groupe est un ensemble muni d'une loi de composition interne<a href="javascript:" title="Lire la suite" class="read-more ms-1">[...]</a></p></div></div><div class="clearfix"></div></div><ul class="pagination justify-content-center public-links "><li class="page-item active"><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 1" class="page-link" title="Abonnez-vous" data-loaded="true">1</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 2" class="page-link" title="Abonnez-vous" data-loaded="true">2</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 3" class="page-link" title="Abonnez-vous" data-loaded="true">3</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 4" class="page-link" title="Abonnez-vous" data-loaded="true">4</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 5" class="page-link" title="Abonnez-vous" data-loaded="true">5</a></li><li class="page-item "><a href="/abonnement" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="page 6" class="page-link" title="Abonnez-vous" data-loaded="true">6</a></li></ul><div class="folio-block"><p class="folio-block-title text-center text-uc"><strong>La suite de cet article est accessible aux abonn茅s</strong></p><ul class="list-unstyled"><li><i class="fa fa-fw fa-check-circle"></i> Des contenus vari茅s, complets et fiables</li><li><i class="fa fa-fw fa-check-circle"></i> Accessible sur tous les 茅crans</li><li><i class="fa fa-fw fa-check-circle"></i> Pas de publicit茅</li></ul><p class="text-center"><a href="/abonnement/" class="btn btn-cta text-uc" id="folio-button" data-matomo-category="Article" data-matomo-name="Folio" data-matomo-value="Block bottom">D茅couvrez nos offres </a></p><p class="text-center text-sm"> D茅j脿 abonn茅 ? <a href="/identification/" id="folio-login">Se connecter</a></p></div><div id="ad_mpuslot" class="pub-container text-center"><div class="pub-placeholder"></div></div><section class="mt-3 mb-5"><i class="anchor" id="A911901_AUTHORS"></i><p class="title-minor">脡crit par</p><ul class="list-group"><li class="list-group-item" data-dc-author="VERLEY, Jean-Luc"><a title="Auteur : Jean-Luc VERLEY" rel="author" href="/auteurs/jean-luc-verley/"><strong>Jean-Luc VERLEY</strong></a> : ma卯tre de conf茅rences honoraire 脿 l'universit茅 de Paris-VII </li></ul></section><section class="mb-5"><div class="text-center mb-3 d-print-none"><a class="d-inline-flex flex-row justify-content-center align-items-center btn btn-light btn-large text-start" title="Carte mentale : ALG脠BRE" href="/carte-mentale/algebre/"><i class="fa fa-sitemap fa-rotate-270 me-1 fs-1 text-universalis"></i><p class="m-0"><strong class="fs-4">Carte mentale</strong><br /> 脡largissez votre recherche dans Universalis </p></a></div></section> <section class="mb-5 article-classification" data-print-label="la classification" data-print-target=".article-classification"> <i class="anchor" id="yA911901_ROOT_LINES"></i> <p class="title-minor">Classification</p> <nav aria-label="Classification 1"> <ol class="breadcrumb"> <li class="breadcrumb-item"> <a title="Classification : Math茅matiques" href="/classification/mathematiques/"> Math茅matiques </a> </li> <li class="breadcrumb-item active" data-dc-subject="Histoire des math茅matiques"> <a title="Classification : Histoire des math茅matiques" href="/classification/mathematiques/histoire-des-mathematiques/"> Histoire des math茅matiques </a> </li> </ol> </nav> <nav aria-label="Classification 2"> <ol class="breadcrumb"> <li class="breadcrumb-item"> <a title="Classification : Math茅matiques" href="/classification/mathematiques/"> Math茅matiques </a> </li> <li class="breadcrumb-item active" data-dc-subject="Alg猫bre"> <a title="Classification : Alg猫bre" href="/classification/mathematiques/algebre/"> Alg猫bre </a> </li> </ol> </nav> <nav aria-label="Classification 3"> <ol class="breadcrumb"> <li class="breadcrumb-item"> <a title="Classification : Math茅matiques" href="/classification/mathematiques/"> Math茅matiques </a> </li> <li class="breadcrumb-item active" data-dc-subject="Topologie"> <a title="Classification : Topologie" href="/classification/mathematiques/topologie/"> Topologie </a> </li> </ol> </nav> </section> <section class="mb-5 d-print-none learn-more"> <i class="anchor" id="yA911901_REFERENCES"></i> <p class="title-minor">Autres r茅f茅rences</p> <ul class="list-group"> <li class="list-group-item"> <h3 class="fs-5 m-0 mt-1"> <a href="/encyclopedie/al-khwarizmi/"> <span class="rg">AL-KHWARIZMI</span> </a> </h3> <ul class="list-inline list-inline-bullet mb-1 text-muted"> <li class="list-inline-item text-sm">脡crit par <span rel="author">Bernard PIRE</span> </li> <li class="list-inline-item text-sm">183 mots</li> </ul> <div class="snippet-content mt-3"> <p> R茅sident de la maison de la Sagesse 脿 Bagdad, le math茅maticien Abu Ja'far Muhammad ibn Musa al-Khwarizmi a particip茅脿 la traduction de nombreux manuscrits scientifiques grecs. Son trait茅 intitul茅 <em>Hisab al-jabr w'al-muqabala</em> est consid茅r茅 comme le premier manuel d'<a data-bs-toggle="popover" data-bs-content="L' alg猫bre au sens moderne, 脿 savoir l'茅tude des structures alg茅briques ind茅pendamment de leurs r茅alisations concr猫tes, ne s'est d茅gag茅e que tr猫s progressivement au cours du xix e 聽si猫cle, en liaison avec le mouvement g茅n茅ral d'axiomatisation de l'ensemble des math茅matiques et la pr茅occupation croissante des math茅maticiens de 芦聽substituer les id茅es au calcul聽禄聽; jusqu'a..." title="Article : ALG脠BRE" href="/encyclopedie/algebre/">alg猫bre...</a></p> </div> </li> <li class="list-group-item"> <h3 class="fs-5 m-0 mt-1"> <a href="/encyclopedie/emil-artin/"> <span class="rg">ARTIN </span><span class="pc">EMIL </span><span class="rm">(1898-1962)</span> </a> </h3> <ul class="list-inline list-inline-bullet mb-1 text-muted"> <li class="list-inline-item text-sm">脡crit par <span rel="author">Jean-Luc VERLEY</span> </li> <li class="list-inline-item text-sm">1 319 mots</li> </ul> <div class="snippet-content mt-3"> <p>On peut consid茅rer Artin comme un des fondateurs de l'<a data-bs-toggle="popover" data-bs-content="L' alg猫bre au sens moderne, 脿 savoir l'茅tude des structures alg茅briques ind茅pendamment de leurs r茅alisations concr猫tes, ne s'est d茅gag茅e que tr猫s progressivement au cours du xix e 聽si猫cle, en liaison avec le mouvement g茅n茅ral d'axiomatisation de l'ensemble des math茅matiques et la pr茅occupation croissante des math茅maticiens de 芦聽substituer les id茅es au calcul聽禄聽; jusqu'a..." title="Article : ALG脠BRE" href="/encyclopedie/algebre/">alg猫bre</a> contemporaine聽; par exemple, de l'aveu de son auteur, le livre <em>Moderne Algebra</em> de Van der Waerden, qui fut l'ouvrage de r茅f茅rence pendant trente ans, est issu de le莽ons profess茅es par Emil Artin et <a data-bs-toggle="popover" data-bs-content="Biographie d'EMMY NOETHER (1882-1935). En quel sens Emmy Noether a-t-elle chang茅 la face de l'alg猫bre聽? D'une science o霉 dominaient les calculs, o霉 l'on discutait 茅quations et syst猫mes, o霉 l'on n'envisageait les probl猫mes que sous leur aspect particulier, elle est parvenue 脿 faire une discipline g茅n茅rale, reposant sur un petit nombre de concepts. Imposant 脿 certains de ces concepts q..." title="Article : EMMY NOETHER (1882-1935)" href="/encyclopedie/emmy-noether/">Emmy Noether</a>. Tant...</p> </div> </li> <li class="list-group-item"> <h3 class="fs-5 m-0 mt-1"> <a href="/encyclopedie/michael-francis-atiyah/"> <span class="rg">ATIYAH </span><span class="pc">MICHAEL FRANCIS </span><span class="rm">(1929-2019)</span> </a> </h3> <ul class="list-inline list-inline-bullet mb-1 text-muted"> <li class="list-inline-item text-sm">脡crit par <span rel="author">Bernard PIRE</span> </li> <li class="list-inline-item text-sm">367 mots</li> </ul> <div class="snippet-content mt-3"> <p> Math茅maticien britannique, laur茅at de la <a data-bs-toggle="popover" data-bs-content="M茅daille des prix internationaux de math茅matiques, qui est, avec le prix Abel (depuis 2003), l'une des plus hautes distinctions dans cette science. C'est selon le d茅sir posthume du math茅maticien canadien John Charles Fields (1863-1932) que le IX e 聽Congr猫s international des math茅maticiens, qui s'est tenu 脿 Zurich en septembre 1932, a enregistr茅 la cr茅ation de deux prix de math茅matiq..." title="Article : M脡DAILLE FIELDS" href="/encyclopedie/medaille-fields/">m茅daille Fields</a> en 1966 pour ses travaux en topologie. N茅 le 22聽avril 1929 脿 Londres, Michael Francis Atiyah fait ses 茅tudes primaires 脿 Khartoum (Soudan), secondaires au Victoria College du Caire et 脿 Alexandrie en 脡gypte, sup茅rieures enfin au Trinity College...</p> </div> </li> <li class="list-group-item"> <h3 class="fs-5 m-0 mt-1"> <a href="/encyclopedie/enrico-betti/"> <span class="rg">BETTI </span><span class="pc">ENRICO </span><span class="rm">(1823-1892)</span> </a> </h3> <ul class="list-inline list-inline-bullet mb-1 text-muted"> <li class="list-inline-item text-sm">脡crit par <span rel="author">Bernard PIRE</span> </li> <li class="list-inline-item text-sm">341 mots</li> </ul> <div class="snippet-content mt-3"> <p> Math茅maticien italien, sp茅cialiste d'<a data-bs-toggle="popover" data-bs-content="L' alg猫bre au sens moderne, 脿 savoir l'茅tude des structures alg茅briques ind茅pendamment de leurs r茅alisations concr猫tes, ne s'est d茅gag茅e que tr猫s progressivement au cours du xix e 聽si猫cle, en liaison avec le mouvement g茅n茅ral d'axiomatisation de l'ensemble des math茅matiques et la pr茅occupation croissante des math茅maticiens de 芦聽substituer les id茅es au calcul聽禄聽; jusqu'a..." title="Article : ALG脠BRE" href="/encyclopedie/algebre/">alg猫bre</a> et de topologie. N茅 le 21聽octobre 1823 脿 Pistoia, en Toscane (Italie), Enrico Betti est tr猫s t么t orphelin de p猫re. Il fait ses 茅tudes 脿 l'universit茅 de Pise, o霉 il enseignera 脿 partir de 1846, et prend part 脿 la guerre d'ind茅pendance de la Toscane...</p> </div> </li> <li class="list-unstyled text-center mt-3"> <a class="btn btn-sm btn-primary" title="Afficher la liste compl猫te des r茅f茅rences" rel="related" href="/index/algebre/"> Afficher les 35 r茅f茅rences <i aria-hidden="true" class="fa fa-fw fa-chevron-right"></i> </a> </li> </ul> </section> <section class="mb-5 d-print-none article-indexes learn-more"> <i class="anchor" id="yA911901_INDEXES"></i> <p class="title-minor">Voir aussi</p> <ul class="list-inline list-inline-bullet"> <li class="list-inline-item m-1"> <a title="Index : VALUATION" href="/index/valuation/"> <span class="rg">VALUATION</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : QUATERNIONS" href="/index/quaternions/"> <span class="rg">QUATERNIONS</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : ID脡AL, math茅matiques" href="/index/ideal-mathematiques/"> <span class="rg">IDÉAL, </span><em class="it">mathématiques</em> </a> </li> <li class="list-inline-item m-1"> <a title="Index : LOCAL ANNEAU" href="/index/anneau-local/"> <span class="rg">LOCAL </span><span class="pc">ANNEAU</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : ALG脠BRES" href="/index/algebres/"> <span class="rg">ALGÈBRES</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : NOMBRES ALG脡BRIQUES CORPS DE" href="/index/corps-de-nombres-algebriques/"> <span class="rg">NOMBRES ALGÉBRIQUES </span><span class="pc">CORPS DE</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : ALG脡BRIQUE ENTIER ou ENTIER RELATIF" href="/index/algebrique-entier-relatif/"> <span class="rg">ALGÉBRIQUE </span><span class="pc">ENTIER </span><span class="rm">ou </span><span class="rg">ENTIER RELATIF</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : ESPACE VECTORIEL" href="/index/espace-vectoriel/"> <span class="rg">ESPACE VECTORIEL</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : STEINITZ ERNST (1871-1928)" href="/index/ernst-steinitz/"> <span class="rg">STEINITZ </span><span class="pc">ERNST </span><span class="rm">(1871-1928)</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : GROUPES FINIS" href="/index/groupes-finis/"> <span class="rg">GROUPES FINIS</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : NOMBRE ID脡AL" href="/index/nombre-ideal/"> <span class="rg">NOMBRE IDÉAL</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : GROUPE TOPOLOGIQUE" href="/index/groupe-topologique/"> <span class="rg">GROUPE TOPOLOGIQUE</span> </a> </li> <li class="list-inline-item m-1"> <a title="Index : FERMAT GRAND TH脡OR脠ME DE" href="/index/grand-theoreme-de-fermat/"> <span class="rg">FERMAT </span><span class="pc">GRAND THÉORÈME DE</span> </a> </li> <li class="list-inline-item m-1"> <a 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