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class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Théorie de Kummer</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 7 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-7" 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">7 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kummer-Theorie" title="Kummer-Theorie – allemand" lang="de" hreflang="de" data-title="Kummer-Theorie" 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/Kummer_theory" title="Kummer theory – anglais" lang="en" hreflang="en" data-title="Kummer theory" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_Kummer" title="Teoría de Kummer – espagnol" lang="es" hreflang="es" data-title="Teoría de Kummer" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_di_Kummer" title="Teoria di Kummer – italien" lang="it" hreflang="it" data-title="Teoria di Kummer" 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/%E3%82%AF%E3%83%B3%E3%83%9E%E3%83%BC%E7%90%86%E8%AB%96" title="クンマー理論 – japonais" lang="ja" hreflang="ja" data-title="クンマー理論" 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-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_de_Kummer" title="Teoria de Kummer – portugais" lang="pt" hreflang="pt" data-title="Teoria de Kummer" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%9A%D1%83%D0%BC%D0%BC%D0%B5%D1%80%D0%B0" title="Теория Куммера – russe" lang="ru" hreflang="ru" data-title="Теория Куммера" data-language-autonym="Русский" data-language-local-name="russe" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata bandeau-article bandeau-niveau-ebauche"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Fichier:Evariste_galois.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Evariste_galois.jpg/27px-Evariste_galois.jpg" decoding="async" width="27" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Evariste_galois.jpg/41px-Evariste_galois.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Evariste_galois.jpg/54px-Evariste_galois.jpg 2x" data-file-width="792" data-file-height="1024" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p><strong class="bandeau-titre">Cet article est une <a href="/wiki/Aide:%C3%89bauche" title="Aide:Ébauche">ébauche</a> concernant l’<a href="/wiki/Alg%C3%A8bre" title="Algèbre">algèbre</a>.</strong> </p><p>Vous pouvez partager vos connaissances en l’améliorant (<b><a href="/wiki/Aide:Comment_modifier_une_page" title="Aide:Comment modifier une page">comment ?</a></b>) selon les recommandations des <a href="/wiki/Projet:Accueil" title="Projet:Accueil">projets correspondants</a>. </p><p>Consultez la liste des <b>tâches à accomplir</b> en <a href="/wiki/Discussion:Th%C3%A9orie_de_Kummer" title="Discussion:Théorie de Kummer">page de discussion</a>. </p> </div></div> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, la <b>théorie de Kummer</b>, ainsi désignée suivant le nom du mathématicien allemand du <abbr class="abbr" title="19ᵉ siècle"><span class="romain">XIX</span><sup style="font-size:72%">e</sup></abbr> siècle <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a>, à la suite de ses travaux sur le <a href="/wiki/Dernier_th%C3%A9or%C3%A8me_de_Fermat" title="Dernier théorème de Fermat">dernier théorème de Fermat</a>, donne une description de certaines <a href="/wiki/Extension_de_corps" title="Extension de corps">extensions d'un corps</a> contenant suffisamment de <a href="/wiki/Racine_de_l%27unit%C3%A9" title="Racine de l'unité">racines de l'unité</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Extension_de_Kummer">Extension de Kummer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9orie_de_Kummer&veaction=edit&section=1" title="Modifier la section : Extension de Kummer" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9orie_de_Kummer&action=edit&section=1" title="Modifier le code source de la section : Extension de Kummer"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Soit un corps commutatif <i>K</i> contenant les racines <i>n</i>-èmes de l'unité, pour <i>n</i> un nombre entier <a href="/wiki/Nombres_premiers_entre_eux" title="Nombres premiers entre eux">premier</a> à la <a href="/wiki/Caract%C3%A9ristique_d%27un_anneau" title="Caractéristique d'un anneau">caractéristique</a> de <i>K</i> si elle est non nulle. Une extension <i>L/K</i> est de Kummer si le corps <i>L</i> est <i>K</i>-engendré par une racine d'un polynôme <i>X<sup>n</sup>-a</i> à coefficients dans <i>K</i>. </p><p>Par exemple, en caractéristique différente de <i>2</i>, l'expression des racines des <a href="/wiki/Trin%C3%B4me_du_second_degr%C3%A9" class="mw-redirect" title="Trinôme du second degré">trinômes du second degré</a> montre que toute <a href="/wiki/Extension_quadratique" title="Extension quadratique">extension quadratique</a> est une extension de Kummer. En revanche, en caractéristique <i>2</i>, il n'y a pas d'extension de Kummer de degré <i>2</i>. </p><p>Si le corps <i>K</i> ne contient aucune racine <i>m</i>-ème de <i>a</i>, pour <i>m>1</i> divisant <i>n</i>, l'extension de Kummer de <i>K</i> donnée par les racines du polynôme <i>X<sup>n</sup>-a</i> est une <a href="/wiki/Extension_galoisienne" class="mw-redirect" title="Extension galoisienne">extension galoisienne</a>, de <a href="/wiki/Groupe_de_Galois" title="Groupe de Galois">groupe de Galois</a> cyclique d'ordre <i>n</i>, dont un générateur <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> est défini par son action sur une racine <i>n</i>-ème de <i>a</i> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma {\sqrt[{n}]{a}}=\zeta _{n}{\sqrt[{n}]{a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>=</mo> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mroot> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma {\sqrt[{n}]{a}}=\zeta _{n}{\sqrt[{n}]{a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cf50c0f5876a024f7fd4f421218a2248c22b7e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.643ex; height:3.009ex;" alt="{\displaystyle \sigma {\sqrt[{n}]{a}}=\zeta _{n}{\sqrt[{n}]{a}},}"></span></dd></dl> <p>où <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 \zeta _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e41a78e5c2df8ff2fab3863a0f10fa7539b973c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.237ex; height:2.509ex;" alt="{\displaystyle \zeta _{n}}"></span> désigne une racine primitive <i>n</i>-ème de l'unité. Sans hypothèse sur les racines de <i>a</i> dans <i>K</i>, l'extension est cyclique d'ordre divisant <i>n</i>. </p><p>Plus généralement, on parle d'extension de Kummer pour un <a href="/wiki/Compositum" title="Compositum">compositum</a> d'un nombre fini d'extensions de Kummer élémentaires telles que décrites ci-dessus, c'est-à-dire obtenues en adjoignant les racines <i>n</i>-èmes d'un nombre fini d'éléments <i>a<sub>i</sub></i> du corps <i>K</i>. Une telle extension est à nouveau galoisienne, et son groupe de Galois est produit direct de <a href="/wiki/Groupe_cyclique" title="Groupe cyclique">groupes cycliques</a> d'ordre divisant <i>n</i>, elle est donc <a href="/wiki/Extension_ab%C3%A9lienne" title="Extension abélienne">abélienne</a>, et son groupe de Galois a un exposant qui divise <i>n</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Théorie_de_Kummer"><span id="Th.C3.A9orie_de_Kummer"></span>Théorie de Kummer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9orie_de_Kummer&veaction=edit&section=2" title="Modifier la section : Théorie de Kummer" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9orie_de_Kummer&action=edit&section=2" title="Modifier le code source de la section : Théorie de Kummer"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La théorie de Kummer traite les réciproques, c'est-à-dire la question : dans quelle mesure les extensions abéliennes d'un corps peuvent-elles être décrites à l'aide de racines <i>n</i>-èmes d'éléments de ce corps ? </p><p>Une réponse est donnée par l'énoncé : si <i>K</i> est un corps contenant les racines <i>n</i>-èmes de l'unité, pour <i>n</i> un entier que ne divise pas la caractéristique de <i>K</i>, et si <i>L</i> est une extension abélienne finie de <i>K</i>, dont le groupe de Galois est d'exposant divisant <i>n</i>, alors il existe des éléments <i>a<sub>1</sub>,...,a<sub>r</sub></i> de <i>K</i> tels 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 L=K({\sqrt[{n}]{a_{1}}},\dots ,{\sqrt[{n}]{a_{r}}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>K</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mroot> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </mroot> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=K({\sqrt[{n}]{a_{1}}},\dots ,{\sqrt[{n}]{a_{r}}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9625f702065b482da8b01d3b6a404c59974ca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.094ex; height:3.176ex;" alt="{\displaystyle L=K({\sqrt[{n}]{a_{1}}},\dots ,{\sqrt[{n}]{a_{r}}})}"></span>. Les éléments <i>a<sub>1</sub></i>, ..., <i>a<sub>r</sub></i> engendrent un sous-groupe abélien <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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span>, d'exposant divisant <i>n</i>, du groupe quotient <i>K<sup>*</sup>/(K<sup>*</sup>)<sup>n</sup></i>. Il y a donc une correspondance entre de tels sous-groupes <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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> et les extensions abéliennes de <i>K</i> de groupe de Galois <i>G</i> d'exposant divisant <i>n</i>, et il existe un <a href="/wiki/Isomorphisme_de_groupes" class="mw-redirect" title="Isomorphisme de groupes">isomorphisme</a> : </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\Delta &\to &Hom(G,\mu _{n})\\a_{i}&\mapsto &(\sigma \mapsto {\frac {\sigma (a_{i})}{a_{i}}})\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi mathvariant="normal">Δ</mi> </mtd> <mtd> <mo stretchy="false">→</mo> </mtd> <mtd> <mi>H</mi> <mi>o</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">↦</mo> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\Delta &\to &Hom(G,\mu _{n})\\a_{i}&\mapsto &(\sigma \mapsto {\frac {\sigma (a_{i})}{a_{i}}})\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9482863470bf9c45d4a29bc5b2ef4162ceeef4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:25.953ex; height:7.843ex;" alt="{\displaystyle {\begin{pmatrix}\Delta &\to &Hom(G,\mu _{n})\\a_{i}&\mapsto &(\sigma \mapsto {\frac {\sigma (a_{i})}{a_{i}}})\end{pmatrix}}}"></span></dd></dl> <p>En notant <i>n<sub>i</sub></i> le plus grand entier divisant <i>n</i> tel que la racine <i>n<sub>i</sub></i>-ème de <i>a</i> soit dans <i>K</i>, il existe donc une famille d'éléments <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 \sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab3208a7d0c634ef720e03ff5a9949e8310edc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.127ex; height:2.009ex;" alt="{\displaystyle \sigma _{i}}"></span> du groupe <i>G</i> qui laissent fixes les <i>a<sub>j</sub></i> pour <i>j</i> différent de <i>i</i>, et agissent sur <i>a<sub>i</sub></i> par multiplication par <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 \zeta _{n}^{n_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta _{n}^{n_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/246694c31c43c8c764696c6e795e9df4cc5fccc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.843ex;" alt="{\displaystyle \zeta _{n}^{n_{i}}}"></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=Th%C3%A9orie_de_Kummer&veaction=edit&section=3" 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=Th%C3%A9orie_de_Kummer&action=edit&section=3" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dans le cas d'une extension abélienne d'ordre <i>p</i>, d'un <a href="/wiki/Corps_commutatif" title="Corps commutatif">corps commutatif</a> de caractéristique <i>p</i>, voir : <a href="/wiki/Th%C3%A9orie_d%27Artin-Schreier" title="Théorie d'Artin-Schreier">théorie d'Artin-Schreier</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Référence"><span id="R.C3.A9f.C3.A9rence"></span>Référence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Th%C3%A9orie_de_Kummer&veaction=edit&section=4" title="Modifier la section : Référence" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Th%C3%A9orie_de_Kummer&action=edit&section=4" title="Modifier le code source de la section : Référence"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="ouvrage" id="Neukirch"><span class="ouvrage" id="Jürgen_Neukirch"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Jürgen <span class="nom_auteur">Neukirch</span></a>, <cite class="italique" lang="en">Algebraic Number Theory</cite> <small>[<a href="/wiki/R%C3%A9f%C3%A9rence:Algebraic_number_theory_(Neukirch)" title="Référence:Algebraic number theory (Neukirch)">détail des éditions</a>]</small><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Number+Theory&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Ffr.wikipedia.org%3ATh%C3%A9orie+de+Kummer"></span></span></span> </p> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail des mathématiques"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/24px-Racine_carr%C3%A9e_bleue.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/36px-Racine_carr%C3%A9e_bleue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Racine_carr%C3%A9e_bleue.svg/48px-Racine_carr%C3%A9e_bleue.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Math%C3%A9matiques" title="Portail:Mathématiques">Portail des mathématiques</a></span> </span></li> <li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer skin-invert-image" typeof="mw:File"><a href="/wiki/Portail:Arithm%C3%A9tique_et_th%C3%A9orie_des_nombres" title="Arithmétique et théorie des nombres"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/34px-Pascal%27s_triangle_5.svg.png" decoding="async" width="34" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/50px-Pascal%27s_triangle_5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Pascal%27s_triangle_5.svg/67px-Pascal%27s_triangle_5.svg.png 2x" data-file-width="540" data-file-height="389" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Arithm%C3%A9tique_et_th%C3%A9orie_des_nombres" title="Portail:Arithmétique et théorie des nombres">Arithmétique et théorie des nombres</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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