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id="toc-Integral_closure_in_algebraic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_closure_in_algebraic_number_theory"> <div class="vector-toc-text"> 2.1 Integral closure in algebraic number theory </div> </a> <ul id="toc-Integral_closure_in_algebraic_number_theory-sublist" class="vector-toc-list"> <li id="toc-Integral_closure_of_integers_in_rationals" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Integral_closure_of_integers_in_rationals"> <div class="vector-toc-text"> 2.1.1 Integral closure of integers in rationals </div> </a> <ul id="toc-Integral_closure_of_integers_in_rationals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quadratic_extensions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quadratic_extensions"> <div class="vector-toc-text"> 2.1.2 Quadratic extensions </div> </a> <ul id="toc-Quadratic_extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Roots_of_unity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Roots_of_unity"> <div class="vector-toc-text"> 2.1.3 Roots of unity </div> </a> <ul id="toc-Roots_of_unity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ring_of_algebraic_integers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ring_of_algebraic_integers"> <div class="vector-toc-text"> 2.1.4 Ring of algebraic integers </div> </a> <ul id="toc-Ring_of_algebraic_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Other"> <div class="vector-toc-text"> 2.1.5 Other </div> </a> <ul id="toc-Other-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Integral_closure_in_algebraic_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_closure_in_algebraic_geometry"> <div class="vector-toc-text"> 2.2 Integral closure in algebraic geometry </div> </a> <ul id="toc-Integral_closure_in_algebraic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integrality_in_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integrality_in_algebra"> <div class="vector-toc-text"> 2.3 Integrality in algebra </div> </a> <ul id="toc-Integrality_in_algebra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalent_definitions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Equivalent_definitions"> <div class="vector-toc-text"> 3 Equivalent definitions </div> </a> <ul id="toc-Equivalent_definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_properties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elementary_properties"> <div class="vector-toc-text"> 4 Elementary properties </div> </a> <button aria-controls="toc-Elementary_properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Elementary properties subsection </button> <ul id="toc-Elementary_properties-sublist" class="vector-toc-list"> <li id="toc-Integral_closure_forms_a_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_closure_forms_a_ring"> <div class="vector-toc-text"> 4.1 Integral closure forms a ring </div> </a> <ul id="toc-Integral_closure_forms_a_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transitivity_of_integrality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transitivity_of_integrality"> <div class="vector-toc-text"> 4.2 Transitivity of integrality </div> </a> <ul id="toc-Transitivity_of_integrality-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_closed_in_fraction_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Integral_closed_in_fraction_field"> <div class="vector-toc-text"> 4.3 Integral closed in fraction field </div> </a> <ul id="toc-Integral_closed_in_fraction_field-sublist" class="vector-toc-list"> <li id="toc-Transitivity_of_integral_closure_with_integrally_closed_domains" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Transitivity_of_integral_closure_with_integrally_closed_domains"> <div class="vector-toc-text"> 4.3.1 Transitivity of integral closure with integrally closed domains </div> </a> <ul id="toc-Transitivity_of_integral_closure_with_integrally_closed_domains-sublist" class="vector-toc-list"> <li id="toc-Transitivity_in_algebraic_number_theory" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Transitivity_in_algebraic_number_theory"> <div class="vector-toc-text"> 4.3.1.1 Transitivity in algebraic number theory </div> </a> <ul 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id="toc-Galois_actions_on_integral_extensions_of_integrally_closed_domains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Galois_actions_on_integral_extensions_of_integrally_closed_domains"> <div class="vector-toc-text"> 5.3 Galois actions on integral extensions of integrally closed domains </div> </a> <ul id="toc-Galois_actions_on_integral_extensions_of_integrally_closed_domains-sublist" class="vector-toc-list"> <li id="toc-Application_to_algebraic_number_theory" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Application_to_algebraic_number_theory"> <div class="vector-toc-text"> 5.3.1 Application to algebraic number theory </div> </a> <ul id="toc-Application_to_algebraic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Remarks_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Remarks_2"> <div class="vector-toc-text"> 5.3.2 Remarks </div> </a> <ul id="toc-Remarks_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Integral_closure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Integral_closure"> <div class="vector-toc-text"> 6 Integral closure </div> </a> <ul id="toc-Integral_closure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conductor" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conductor"> <div class="vector-toc-text"> 7 Conductor </div> </a> <ul id="toc-Conductor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finiteness_of_integral_closure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Finiteness_of_integral_closure"> <div class="vector-toc-text"> 8 Finiteness of integral closure </div> </a> <ul id="toc-Finiteness_of_integral_closure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noether's_normalization_lemma" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Noether's_normalization_lemma"> <div class="vector-toc-text"> 9 Noether's normalization lemma </div> </a> <ul id="toc-Noether's_normalization_lemma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_morphisms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Integral_morphisms"> <div class="vector-toc-text"> 10 Integral morphisms </div> </a> <ul id="toc-Integral_morphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Absolute_integral_closure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Absolute_integral_closure"> <div class="vector-toc-text"> 11 Absolute integral closure </div> </a> <ul id="toc-Absolute_integral_closure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 14 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 15 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" 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class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Entjera_elemento" title="Entjera elemento – Esperanto" lang="eo" hreflang="eo" data-title="Entjera elemento" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89l%C3%A9ment_entier" title="Élément entier – French" lang="fr" hreflang="fr" data-title="Élément entier" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EC%88%98%EC%A0%81_%EC%9B%90%EC%86%8C" title="정수적 원소 – Korean" lang="ko" hreflang="ko" data-title="정수적 원소" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Estensione_intera" title="Estensione intera – Italian" lang="it" hreflang="it" data-title="Estensione intera" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geheel_element" title="Geheel element – Dutch" lang="nl" hreflang="nl" data-title="Geheel element" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B4%E6%8B%A1%E5%A4%A7" title="整拡大 – Japanese" lang="ja" hreflang="ja" data-title="整拡大" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Element_ca%C5%82kowity" title="Element całkowity – Polish" lang="pl" hreflang="pl" data-title="Element całkowity" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A6%D0%B5%D0%BB%D1%8B%D0%B9_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82" title="Целый элемент – Russian" lang="ru" hreflang="ru" data-title="Целый элемент" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kokonaissulkeuma" title="Kokonaissulkeuma – Finnish" lang="fi" hreflang="fi" data-title="Kokonaissulkeuma" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D1%96%D0%BB%D0%B5_%D1%80%D0%BE%D0%B7%D1%88%D0%B8%D1%80%D0%B5%D0%BD%D0%BD%D1%8F_%D0%BA%D1%96%D0%BB%D1%8C%D1%86%D1%8F" title="Ціле розширення кільця – Ukrainian" lang="uk" hreflang="uk" data-title="Ціле розширення кільця" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%95%B4%E6%80%A7" title="整性 – Chinese" lang="zh" hreflang="zh" data-title="整性" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1493740#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical element</div> In <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, an element b of a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> B is said to be integral over a <a href="/wiki/Subring" title="Subring">subring</a> A of B if b is a <a href="/wiki/Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">root</a> of some <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> over A.<a href="#cite_note-1">[1]</a> If A, B are <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, then the notions of "integral over" and of an "integral extension" are precisely "<a href="/wiki/Algebraic_element" title="Algebraic element">algebraic</a> over" and "<a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic extensions</a>" in <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a> (since the root of any <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> is the root of a monic polynomial). The case of greatest interest in <a href="/wiki/Number_theory" title="Number theory">number theory</a> is that of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> integral over Z (e.g., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a65e41a5c0369e908cf26a2452046f19bab946d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.805ex; height:2.343ex;" alt="{\displaystyle 1+i}">); in this context, the integral elements are usually called <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>. The algebraic integers in a finite <a href="/wiki/Field_extension" title="Field extension">extension field</a> k of the <a href="/wiki/Rational_number" title="Rational number">rationals</a> Q form a subring of k, called the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> of k, a central object of study in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. In this article, the term <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> will be understood to mean commutative ring with a multiplicative identity. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> be a ring and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subset B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊂</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subset B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/010e98bb4c817357e3ef7e8fa7fbe2385b2aec6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.606ex; height:2.176ex;" alt="{\displaystyle A\subset B}"> be a subring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eccf5bca7cdc1fa4439af2d31831db6bde00473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}"> An element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is said to be integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> if for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc38ec6af7dd11fdc9baa67365f23906d76da4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.302ex; height:2.509ex;" alt="{\displaystyle n\geq 1,}"> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\ a_{1},\ \dots ,\ a_{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},\ a_{1},\ \dots ,\ a_{n-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7e740e4ffd6cfa7ed9338f5e0ea6b54cd9e2a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.458ex; height:2.009ex;" alt="{\displaystyle a_{0},\ a_{1},\ \dots ,\ a_{n-1}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>b</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2f1f420c5e23e92325debedb373cab8093828d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.64ex; height:3.009ex;" alt="{\displaystyle b^{n}+a_{n-1}b^{n-1}+\cdots +a_{1}b+a_{0}=0.}"> The set of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> that are integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is called the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eccf5bca7cdc1fa4439af2d31831db6bde00473" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle B.}"> The integral closure of any subring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is, itself, a subring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> and contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a71bf21ad35b8fe05555041d54d1e17eeb0f490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.39ex; height:2.176ex;" alt="{\displaystyle A.}"> If every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2746026864cc5896e3e52443a1c917be2df9d8ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.39ex; height:2.509ex;" alt="{\displaystyle A,}"> then we say that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, or equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is an integral extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a71bf21ad35b8fe05555041d54d1e17eeb0f490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.39ex; height:2.176ex;" alt="{\displaystyle A.}"> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=2" title="Edit section: Examples">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Integral_closure_in_algebraic_number_theory">Integral closure in algebraic number theory</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=3" title="Edit section: Integral closure in algebraic number theory">edit</a>]</div> There are many examples of integral closure which can be found in algebraic number theory since it is fundamental for defining the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> for an <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic field extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K/\mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K/\mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce6782a2935122dafec2947d1d7ba168be4399ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.037ex; height:2.843ex;" alt="{\displaystyle K/\mathbb {Q} }"> (or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/\mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/\mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ccef7108c2f7862d7c7703ba7e74e62c1af9792" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.613ex; height:3.009ex;" alt="{\displaystyle L/\mathbb {Q} _{p}}">). <div class="mw-heading mw-heading4"><h4 id="Integral_closure_of_integers_in_rationals">Integral closure of integers in rationals</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=4" title="Edit section: Integral closure of integers in rationals">edit</a>]</div> <a href="/wiki/Integer" title="Integer">Integers</a> are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q. <div class="mw-heading mw-heading4"><h4 id="Quadratic_extensions">Quadratic extensions</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=5" title="Edit section: Quadratic extensions">edit</a>]</div> The <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> are the complex numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {-1}},\,a,b\in \mathbf {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {-1}},\,a,b\in \mathbf {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fabda3e2107c77efffe55ffa48dfa36ca6958002" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.131ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {-1}},\,a,b\in \mathbf {Z} }">, and are integral over Z. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Z} [{\sqrt {-1}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Z} [{\sqrt {-1}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00fbb29cc59061c5429a4ce2557015fcb17790d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:3.009ex;" alt="{\displaystyle \mathbf {Z} [{\sqrt {-1}}]}"> is then the integral closure of Z in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ({\sqrt {-1}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ({\sqrt {-1}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63720d74968afc59ad09dfad31c1eec6454248d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.724ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} ({\sqrt {-1}})}">. Typically this ring is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbb {Q} [i]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbb {Q} [i]}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad9a7d6c23fc0eb5e603476f31926e3486301a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.843ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbb {Q} [i]}}">. The integral closure of Z in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Q} ({\sqrt {5}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Q} ({\sqrt {5}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef808cb113072f209403c6264016a997e67e5ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.916ex; height:3.009ex;" alt="{\displaystyle \mathbf {Q} ({\sqrt {5}})}"> is the ring <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbb {Q} [{\sqrt {5}}]}=\mathbb {Z} \!\left[{\frac {1+{\sqrt {5}}}{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbb {Q} [{\sqrt {5}}]}=\mathbb {Z} \!\left[{\frac {1+{\sqrt {5}}}{2}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d17c9648aabf9c1612458ebd294172d85c01aa1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.507ex; height:6.509ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbb {Q} [{\sqrt {5}}]}=\mathbb {Z} \!\left[{\frac {1+{\sqrt {5}}}{2}}\right]}"></dd></dl> This example and the previous one are examples of <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integers</a>. The integral closure of a quadratic extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ({\sqrt {d}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ({\sqrt {d}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9757d116555dd605228d352039f5f491d2967c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.769ex; height:3.176ex;" alt="{\displaystyle \mathbb {Q} ({\sqrt {d}})}"> can be found by constructing the <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> of an arbitrary element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b{\sqrt {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {d}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6c46254eee2978b88135fbd1d75e89d836dbcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.22ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {d}}}"> and finding number-theoretic criterion for the polynomial to have integral coefficients. This analysis can be found in the <a href="/wiki/Quadratic_integer#Determining_the_ring_of_integers" title="Quadratic integer">quadratic extensions article</a>. <div class="mw-heading mw-heading4"><h4 id="Roots_of_unity">Roots of unity</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=6" title="Edit section: Roots of unity">edit</a>]</div> Let ζ be a <a href="/wiki/Root_of_unity" title="Root of unity">root of unity</a>. Then the integral closure of Z in the <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a> Q(ζ) is Z[ζ].<a href="#cite_note-2">[2]</a> This can be found by using the <a href="/wiki/Minimal_polynomial_(field_theory)" title="Minimal polynomial (field theory)">minimal polynomial</a> and using <a href="/wiki/Eisenstein%27s_criterion" title="Eisenstein's criterion">Eisenstein's criterion</a>. <div class="mw-heading mw-heading4"><h4 id="Ring_of_algebraic_integers">Ring of algebraic integers</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=7" title="Edit section: Ring of algebraic integers">edit</a>]</div> The integral closure of Z in the field of complex numbers C, or the algebraic closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbb {Q} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbb {Q} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/377a8814b1ca454c488e409813988dd5dd906148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.923ex; height:3.343ex;" alt="{\displaystyle {\overline {\mathbb {Q} }}}"> is called the ring of <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>. <div class="mw-heading mw-heading4"><h4 id="Other">Other</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=8" title="Edit section: Other">edit</a>]</div> The <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a>, <a href="/wiki/Nilpotent_element" class="mw-redirect" title="Nilpotent element">nilpotent elements</a> and <a href="/wiki/Idempotent_(ring_theory)" title="Idempotent (ring theory)">idempotent elements</a> in any ring are integral over Z. <div class="mw-heading mw-heading3"><h3 id="Integral_closure_in_algebraic_geometry">Integral closure in algebraic geometry</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=9" title="Edit section: Integral closure in algebraic geometry">edit</a>]</div> In <a href="/wiki/Geometry" title="Geometry">geometry</a>, integral closure is closely related with <a href="/wiki/Noether_normalization_lemma" title="Noether normalization lemma">normalization</a> and <a href="/wiki/Normal_scheme" title="Normal scheme">normal schemes</a>. It is the first step in <a href="/wiki/Resolution_of_singularities" title="Resolution of singularities">resolution of singularities</a> since it gives a process for resolving singularities of codimension 1. <ul><li>For example, the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [x,y,z]/(xy)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x,y,z]/(xy)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12b201390b6601733f40f44075ab2aebe4893f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.07ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [x,y,z]/(xy)}"> is the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [x,z]\times \mathbb {C} [y,z]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x,z]\times \mathbb {C} [y,z]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43461d669ec03ec6912a8958880dea89e8ca8f56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.513ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [x,z]\times \mathbb {C} [y,z]}"> since geometrically, the first ring corresponds to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xz}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97005bee6e83614cf6ce64d4e68e5ab2ac280709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.418ex; height:1.676ex;" alt="{\displaystyle xz}">-plane unioned with the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle yz}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9c7cbe4344a9d72acfc98a97e2b3ec44bb48d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.244ex; height:2.009ex;" alt="{\displaystyle yz}">-plane. They have a codimension 1 singularity along the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">-axis where they intersect.</li> <li>Let a <a href="/wiki/Finite_group" title="Finite group">finite group</a> G <a href="/wiki/Group_action" title="Group action">act</a> on a ring A. Then A is integral over AG, the set of elements fixed by G; see <a href="/wiki/Fixed-point_subring" title="Fixed-point subring">Ring of invariants</a>.</li> <li>Let R be a ring and u a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> in a ring containing R. Then<a href="#cite_note-3">[3]</a></li></ul> <ol><li>u−1 is integral over R <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> u−1 ∈ R[u].</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[u]\cap R[u^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>u</mi> <mo stretchy="false">]</mo> <mo>∩</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[u]\cap R[u^{-1}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6984e3faced97858fbcae9e61b33a063b7e67ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.69ex; height:3.176ex;" alt="{\displaystyle R[u]\cap R[u^{-1}]}"> is integral over R.</li> <li>The integral closure of the <a href="/wiki/Homogeneous_coordinate_ring" title="Homogeneous coordinate ring">homogeneous coordinate ring</a> of a normal <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> X is the <a href="/wiki/Ring_of_sections" class="mw-redirect" title="Ring of sections">ring of sections</a><a href="#cite_note-4">[4]</a></li></ol> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{n\geq 0}\operatorname {H} ^{0}(X,{\mathcal {O}}_{X}(n)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigoplus _{n\geq 0}\operatorname {H} ^{0}(X,{\mathcal {O}}_{X}(n)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c656e076c7b93605aa325c2beb1bf438532121b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.851ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{n\geq 0}\operatorname {H} ^{0}(X,{\mathcal {O}}_{X}(n)).}"></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Integrality_in_algebra">Integrality in algebra</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=10" title="Edit section: Integrality in algebra">edit</a>]</div> <ul><li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {k}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e029b7ba61d598ed35f8d899d3c8ccd39166429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.326ex; height:3.009ex;" alt="{\displaystyle {\overline {k}}}"> is an <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of a field k, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {k}}[x_{1},\dots ,x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {k}}[x_{1},\dots ,x_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8c594eb099b27991c95d80f2169783616bf7394" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.73ex; height:3.509ex;" alt="{\displaystyle {\overline {k}}[x_{1},\dots ,x_{n}]}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{1},\dots ,x_{n}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[x_{1},\dots ,x_{n}].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb94059c60c6f3fcb963b4f1f233ba297117e9d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.262ex; height:2.843ex;" alt="{\displaystyle k[x_{1},\dots ,x_{n}].}"></li> <li>The integral closure of C[[x]] in a finite extension of C((x)) is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} [[x^{1/n}]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} [[x^{1/n}]]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca30dd3f15c8e69abe3c0e5580f6451d2ca2a485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.711ex; height:3.343ex;" alt="{\displaystyle \mathbf {C} [[x^{1/n}]]}"> (cf. <a href="/wiki/Puiseux_series" title="Puiseux series">Puiseux series</a>)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</li></ul> <div class="mw-heading mw-heading2"><h2 id="Equivalent_definitions">Equivalent definitions</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=11" title="Edit section: Equivalent definitions">edit</a>]</div> Let B be a ring, and let A be a subring of B. Given an element b in B, the following conditions are equivalent: <dl><dd>(i) b is integral over A;</dd> <dd>(ii) the subring A[b] of B generated by A and b is a <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated A-module</a>;</dd> <dd>(iii) there exists a subring C of B containing A[b] and which is a finitely generated A-module;</dd> <dd>(iv) there exists a <a href="/wiki/Faithful_module" class="mw-redirect" title="Faithful module">faithful</a> A[b]-module M such that M is finitely generated as an A-module.</dd></dl> The usual <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> of this uses the following variant of the <a href="/wiki/Cayley%E2%80%93Hamilton_theorem" title="Cayley–Hamilton theorem">Cayley–Hamilton theorem</a> on <a href="/wiki/Determinant" title="Determinant">determinants</a>: <dl><dd>Theorem Let u be an <a href="/wiki/Endomorphism" title="Endomorphism">endomorphism</a> of an A-module M generated by n elements and I an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of A such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(M)\subset IM}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>⊂</mo> <mi>I</mi> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(M)\subset IM}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c17ba16ad1fcaa1e9ab003b6b6821d4527cd5094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.294ex; height:2.843ex;" alt="{\displaystyle u(M)\subset IM}">. Then there is a relation: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u^{n}+a_{1}u^{n-1}+\cdots +a_{n-1}u+a_{n}=0,\,a_{i}\in I^{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>u</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{n}+a_{1}u^{n-1}+\cdots +a_{n-1}u+a_{n}=0,\,a_{i}\in I^{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f19cd814df169fe8dd8cf7708a439b7b5822fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:45.108ex; height:3.009ex;" alt="{\displaystyle u^{n}+a_{1}u^{n-1}+\cdots +a_{n-1}u+a_{n}=0,\,a_{i}\in I^{i}.}"></dd></dl></dd></dl> This theorem (with I = A and u multiplication by b) gives (iv) ⇒ (i) and the rest is easy. Coincidentally, <a href="/wiki/Nakayama%27s_lemma" title="Nakayama's lemma">Nakayama's lemma</a> is also an immediate consequence of this theorem. <div class="mw-heading mw-heading2"><h2 id="Elementary_properties">Elementary properties</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=12" title="Edit section: Elementary properties">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Integral_closure_forms_a_ring">Integral closure forms a ring</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=13" title="Edit section: Integral closure forms a ring">edit</a>]</div> It follows from the above four equivalent statements that the set of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> that are integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> forms a subring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. (Proof: If x, y are elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> that are integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y,xy,-x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mi>y</mi> <mo>,</mo> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y,xy,-x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8ba79985d6ea99893651a87bcb400b9d030fb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.016ex; height:2.343ex;" alt="{\displaystyle x+y,xy,-x}"> are integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> since they stabilize <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A[x]A[y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mi>A</mi> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[x]A[y]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54b9c2014ba5f2c3f49fd79d70a48072b4815b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.559ex; height:2.843ex;" alt="{\displaystyle A[x]A[y]}">, which is a finitely generated module over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and is annihilated only by zero.)<a href="#cite_note-5">[5]</a> This ring is called the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. <div class="mw-heading mw-heading3"><h3 id="Transitivity_of_integrality">Transitivity of integrality</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=14" title="Edit section: Transitivity of integrality">edit</a>]</div> Another consequence of the above equivalence is that "integrality" is <a href="/wiki/Transitive_relation" title="Transitive relation">transitive</a>, in the following sense. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> be a ring containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/447d3982c94c23d6b6d01c90da81a6125aa26567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.614ex; height:2.176ex;" alt="{\displaystyle c\in C}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is itself integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> is also integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. <div class="mw-heading mw-heading3"><h3 id="Integral_closed_in_fraction_field">Integral closed in fraction field</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=15" title="Edit section: Integral closed in fraction field">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> happens to be the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, then A is said to be integrally closed in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is the <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">total ring of fractions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, (e.g., the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>), then one sometimes drops the qualification "in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">" and simply says "integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">" and "<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">integrally closed</a>."<a href="#cite_note-6">[6]</a> For example, the ring of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> is integrally closed in the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}">. <div class="mw-heading mw-heading4"><h4 id="Transitivity_of_integral_closure_with_integrally_closed_domains">Transitivity of integral closure with integrally closed domains</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=16" title="Edit section: Transitivity of integral closure with integrally closed domains">edit</a>]</div> Let A be an integral domain with the field of fractions K and A' the integral closure of A in an <a href="/wiki/Algebraic_extension" title="Algebraic extension">algebraic field extension</a> L of K. Then the field of fractions of A' is L. In particular, A' is an <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">integrally closed domain</a>. <div class="mw-heading mw-heading5"><h5 id="Transitivity_in_algebraic_number_theory">Transitivity in algebraic number theory</h5>[<a href="/w/index.php?title=Integral_element&action=edit&section=17" title="Edit section: Transitivity in algebraic number theory">edit</a>]</div> This situation is applicable in algebraic number theory when relating the ring of integers and a field extension. In particular, given a field extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0381945929156997b99bb43e5b7067d18c9a84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}"> the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{K}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/180e5aded86734744a16ec9c668976b5dbafd4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.543ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{K}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> is the ring of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/170fbb174bb98c93172809f50338f114bbd1163a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.201ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{L}}">. <div class="mw-heading mw-heading4"><h4 id="Remarks">Remarks</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=18" title="Edit section: Remarks">edit</a>]</div> Note that transitivity of integrality above implies that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is a union (equivalently an <a href="/wiki/Inductive_limit" class="mw-redirect" title="Inductive limit">inductive limit</a>) of subrings that are finitely generated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">-modules. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is <a href="/wiki/Noetherian_ring" title="Noetherian ring">noetherian</a>, transitivity of integrality can be weakened to the statement: <dl><dd>There exists a finitely generated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">-submodule of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> that contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A[b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca79decb9d0d5e08ac9f77f713e8df03aad88d70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.034ex; height:2.843ex;" alt="{\displaystyle A[b]}">.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Relation_with_finiteness_conditions">Relation with finiteness conditions</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=19" title="Edit section: Relation with finiteness conditions">edit</a>]</div> Finally, the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> be a subring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> can be modified a bit. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}"> is a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a>, then one says <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is integral if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2711c647e5397c7016ee21bbcea53565480bd5e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.831ex; height:2.843ex;" alt="{\displaystyle f(A)}">. In the same way one says <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is finite (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> finitely generated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">-module) or of finite type (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> finitely generated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">-<a href="/wiki/Algebra_over_a_ring" class="mw-redirect" title="Algebra over a ring">algebra</a>). In this viewpoint, one has that <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is finite if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> is integral and of finite type.</dd></dl> Or more explicitly, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is a finitely generated <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">-module if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is generated as an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">-algebra by a finite number of elements integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Integral_extensions">Integral extensions</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=20" title="Edit section: Integral extensions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Cohen-Seidenberg_theorems">Cohen-Seidenberg theorems</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=21" title="Edit section: Cohen-Seidenberg theorems">edit</a>]</div> An integral extension A ⊆ B has the <a href="/wiki/Going_up_and_going_down" title="Going up and going down">going-up property</a>, the <a href="/wiki/Lying_over" class="mw-redirect" title="Lying over">lying over</a> property, and the <a href="/wiki/Going_up_and_going_down#Lying_over_and_incomparability" title="Going up and going down">incomparability</a> property (<a href="/wiki/Going_up_and_going_down" title="Going up and going down">Cohen–Seidenberg theorems</a>). Explicitly, given a chain of <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{1}\subset \cdots \subset {\mathfrak {p}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{1}\subset \cdots \subset {\mathfrak {p}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3b4d68bcc4f2b54660bb4c1870524a55978948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.518ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{1}\subset \cdots \subset {\mathfrak {p}}_{n}}"> in A there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}'_{1}\subset \cdots \subset {\mathfrak {p}}'_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}'_{1}\subset \cdots \subset {\mathfrak {p}}'_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec9b73c9c499acfa77a96dd083b88df90e3091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.518ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}'_{1}\subset \cdots \subset {\mathfrak {p}}'_{n}}"> in B with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{i}={\mathfrak {p}}'_{i}\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{i}={\mathfrak {p}}'_{i}\cap A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cdcfadb404c0b28e9eb9cbda71945b3b1268cad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.348ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}_{i}={\mathfrak {p}}'_{i}\cap A}"> (going-up and lying over) and two distinct prime ideals with inclusion relation cannot contract to the same prime ideal (incomparability). In particular, the <a href="/wiki/Krull_dimension" title="Krull dimension">Krull dimensions</a> of A and B are the same. Furthermore, if A is an integrally closed domain, then the going-down holds (see below). In general, the going-up implies the lying-over.<a href="#cite_note-7">[7]</a> Thus, in the below, we simply say the "going-up" to mean "going-up" and "lying-over". When A, B are domains such that B is integral over A, A is a field if and only if B is a field. As a <a href="/wiki/Corollary" title="Corollary">corollary</a>, one has: given a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b0448036d3f2e513dd80d94095c45d81510269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.137ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}}"> of B, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b0448036d3f2e513dd80d94095c45d81510269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.137ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}}"> is a <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> of B if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}\cap A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc3fff6b2078ac61626a79a5229f18f60dc9e40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {q}}\cap A}"> is a maximal ideal of A. Another corollary: if L/K is an algebraic extension, then any subring of L containing K is a field. <div class="mw-heading mw-heading4"><h4 id="Applications">Applications</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=22" title="Edit section: Applications">edit</a>]</div> Let B be a ring that is integral over a subring A and k an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb67e30890ecf47d83c9dfc9e35f727b648fc5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.784ex; height:2.509ex;" alt="{\displaystyle f:A\to k}"> is a homomorphism, then f extends to a homomorphism B → k.<a href="#cite_note-8">[8]</a> This follows from the going-up. <div class="mw-heading mw-heading4"><h4 id="Geometric_interpretation_of_going-up">Geometric interpretation of going-up</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=23" title="Edit section: Geometric interpretation of going-up">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:A\to B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:A\to B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20040a52d9391f2fe271f0aaa300bf7887a0c7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.337ex; height:2.509ex;" alt="{\displaystyle f:A\to B}"> be an integral extension of rings. Then the induced map <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}f^{\#}:\operatorname {Spec} B\to \operatorname {Spec} A\\p\mapsto f^{-1}(p)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">#</mi> </mrow> </msup> <mo>:</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">↦</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}f^{\#}:\operatorname {Spec} B\to \operatorname {Spec} A\\p\mapsto f^{-1}(p)\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e684028d83ecdb12552f490333765af2c98444c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.548ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}f^{\#}:\operatorname {Spec} B\to \operatorname {Spec} A\\p\mapsto f^{-1}(p)\end{cases}}}"></dd></dl> is a <a href="/wiki/Closed_map" class="mw-redirect" title="Closed map">closed map</a>; in fact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\#}(V(I))=V(f^{-1}(I))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">#</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\#}(V(I))=V(f^{-1}(I))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e13c7f01ca6cdf5ba4fe5aa813bc347c0559841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.829ex; height:3.176ex;" alt="{\displaystyle f^{\#}(V(I))=V(f^{-1}(I))}"> for any ideal I and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{\#}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">#</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{\#}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7226c6398081c26bdb12079d3c6a1a094f848e02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.922ex; height:3.009ex;" alt="{\displaystyle f^{\#}}"> is <a href="/wiki/Surjective_map" class="mw-redirect" title="Surjective map">surjective</a> if f is <a href="/wiki/Injective_map" class="mw-redirect" title="Injective map">injective</a>. This is a geometric interpretation of the going-up. <div class="mw-heading mw-heading4"><h4 id="Geometric_interpretation_of_integral_extensions">Geometric interpretation of integral extensions</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=24" title="Edit section: Geometric interpretation of integral extensions">edit</a>]</div> Let B be a ring and A a subring that is a noetherian integrally closed domain (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad53059b6a9d6c309ab164aa4eaed021844b5ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.78ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} A}"> is a <a href="/wiki/Normal_scheme" title="Normal scheme">normal scheme</a>). If B is integral over A, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e33d3fb1c68e0024bec7edf94421bb48ac78fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.195ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> is <a href="/w/index.php?title=Submersion_(algebra)&action=edit&redlink=1" class="new" title="Submersion (algebra) (page does not exist)">submersive</a>; i.e., the <a href="/wiki/Topological_space" title="Topological space">topology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad53059b6a9d6c309ab164aa4eaed021844b5ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.78ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} A}"> is the <a href="/wiki/Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a>.<a href="#cite_note-9">[9]</a> The proof uses the notion of <a href="/wiki/Constructible_set_(topology)" title="Constructible set (topology)">constructible sets</a>. (See also: <a href="/wiki/Torsor_(algebraic_geometry)" title="Torsor (algebraic geometry)">Torsor (algebraic geometry)</a>.) <div class="mw-heading mw-heading3"><h3 id="Integrality,_base-change,_universally-closed,_and_geometry">Integrality, base-change, universally-closed, and geometry</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=25" title="Edit section: Integrality, base-change, universally-closed, and geometry">edit</a>]</div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\otimes _{A}R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\otimes _{A}R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb64c412d97e29b1d37564f9f47c68a6a53e18b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.833ex; height:2.509ex;" alt="{\displaystyle B\otimes _{A}R}"> is integral over R for any A-algebra R.<a href="#cite_note-10">[10]</a> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} (B\otimes _{A}R)\to \operatorname {Spec} R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} (B\otimes _{A}R)\to \operatorname {Spec} R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d52e4c2fc7dab53bd333b2787f8d0678563e6cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.707ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spec} (B\otimes _{A}R)\to \operatorname {Spec} R}"> is closed; i.e., the integral extension induces a "universally closed" map. This leads to a geometric characterization of integral extension. Namely, let B be a ring with only finitely many <a href="/wiki/Minimal_prime_ideal" title="Minimal prime ideal">minimal prime ideals</a> (e.g., integral domain or noetherian ring). Then B is integral over a (subring) A if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} (B\otimes _{A}R)\to \operatorname {Spec} R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} (B\otimes _{A}R)\to \operatorname {Spec} R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d52e4c2fc7dab53bd333b2787f8d0678563e6cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.707ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spec} (B\otimes _{A}R)\to \operatorname {Spec} R}"> is closed for any A-algebra R.<a href="#cite_note-11">[11]</a> In particular, every proper map is universally closed.<a href="#cite_note-12">[12]</a> <div class="mw-heading mw-heading3"><h3 id="Galois_actions_on_integral_extensions_of_integrally_closed_domains">Galois actions on integral extensions of integrally closed domains</h3>[<a href="/w/index.php?title=Integral_element&action=edit&section=26" title="Edit section: Galois actions on integral extensions of integrally closed domains">edit</a>]</div> <dl><dd>Proposition. Let A be an integrally closed domain with the field of fractions K, L a finite <a href="/wiki/Normal_extension" title="Normal extension">normal extension</a> of K, B the integral closure of A in L. Then the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\operatorname {Gal} (L/K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>Gal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\operatorname {Gal} (L/K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90a7ad81620311f0ad3b1e0666c6e867c992981b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.179ex; height:2.843ex;" alt="{\displaystyle G=\operatorname {Gal} (L/K)}"> acts <a href="/wiki/Group_action#Remarkable_properties_of_actions" title="Group action">transitively</a> on each fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e33d3fb1c68e0024bec7edf94421bb48ac78fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.195ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}">.</dd></dl> Proof. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{2}\neq \sigma ({\mathfrak {p}}_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≠</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{2}\neq \sigma ({\mathfrak {p}}_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d488d305970943bca9703ee76b7b3c861b18f14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.671ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}_{2}\neq \sigma ({\mathfrak {p}}_{1})}"> for any <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><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 }"> in G. Then, by <a href="/wiki/Prime_avoidance" class="mw-redirect" title="Prime avoidance">prime avoidance</a>, there is an element x in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be55dd319953dfaeb676e93d587eda2d8908cc25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.217ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{2}}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (x)\not \in {\mathfrak {p}}_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∉</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (x)\not \in {\mathfrak {p}}_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5edc6c5e4ec79a7417658dbce923a14441d953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.526ex; height:2.843ex;" alt="{\displaystyle \sigma (x)\not \in {\mathfrak {p}}_{1}}"> for any <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><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 }">. G fixes the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\prod \nolimits _{\sigma }\sigma (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <msub> <mo movablelimits="false">∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msub> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=\prod \nolimits _{\sigma }\sigma (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfb38e024a62efec5d8e9289f10f6d825f24e43a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.252ex; height:3.843ex;" alt="{\displaystyle y=\prod \nolimits _{\sigma }\sigma (x)}"> and thus y is <a href="/wiki/Purely_inseparable" class="mw-redirect" title="Purely inseparable">purely inseparable</a> over K. Then some power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0f72fae519109c04aab6fd2d7b546acf4ab256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.159ex; height:2.676ex;" alt="{\displaystyle y^{e}}"> belongs to K; since A is integrally closed we have: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{e}\in A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>∈</mo> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{e}\in A.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ff7bdd7423118d0618c9605e7f9802674bc069b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.389ex; height:2.676ex;" alt="{\displaystyle y^{e}\in A.}"> Thus, we found <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0f72fae519109c04aab6fd2d7b546acf4ab256" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.159ex; height:2.676ex;" alt="{\displaystyle y^{e}}"> is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{2}\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{2}\cap A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b41df0116d88760d7274fe8c05f66d1c96a317f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.542ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}_{2}\cap A}"> but not in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{1}\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{1}\cap A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd1de85037681f348553258153d62356cbe20ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.542ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}_{1}\cap A}">; i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{1}\cap A\neq {\mathfrak {p}}_{2}\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∩</mo> <mi>A</mi> <mo>≠</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{1}\cap A\neq {\mathfrak {p}}_{2}\cap A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b418fa6fc777acc410faec4e3db327cbc21a90a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.183ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}_{1}\cap A\neq {\mathfrak {p}}_{2}\cap A}">. <div class="mw-heading mw-heading4"><h4 id="Application_to_algebraic_number_theory">Application to algebraic number theory</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=27" title="Edit section: Application to algebraic number theory">edit</a>]</div> The Galois group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Gal} (L/K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Gal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Gal} (L/K)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ffa68a2cbf93db0330bd5a20bf0e74305d53b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.254ex; height:2.843ex;" alt="{\displaystyle \operatorname {Gal} (L/K)}"> then acts on all of the prime ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}_{1},\ldots ,{\mathfrak {q}}_{k}\in {\text{Spec}}({\mathcal {O}}_{L})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}_{1},\ldots ,{\mathfrak {q}}_{k}\in {\text{Spec}}({\mathcal {O}}_{L})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc56bf5dccd838cdd31e31c287df1180ac424d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.096ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {q}}_{1},\ldots ,{\mathfrak {q}}_{k}\in {\text{Spec}}({\mathcal {O}}_{L})}"> lying over a fixed prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Spec</mtext> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da65e1a5261a55b04d68f2fe2b5b77f290cc263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.005ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}\in {\text{Spec}}({\mathcal {O}}_{K})}">.<a href="#cite_note-13">[13]</a> That is, if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}={\mathfrak {q}}_{1}^{e_{1}}\cdots {\mathfrak {q}}_{k}^{e_{k}}\subset {\mathcal {O}}_{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>⋯</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msubsup> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}={\mathfrak {q}}_{1}^{e_{1}}\cdots {\mathfrak {q}}_{k}^{e_{k}}\subset {\mathcal {O}}_{L}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a63c0aeeba2315df29133243a571d6511e840e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.02ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {p}}={\mathfrak {q}}_{1}^{e_{1}}\cdots {\mathfrak {q}}_{k}^{e_{k}}\subset {\mathcal {O}}_{L}}"></dd></dl> then there is a Galois action on the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{\mathfrak {p}}=\{{\mathfrak {q}}_{1},\ldots ,{\mathfrak {q}}_{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{\mathfrak {p}}=\{{\mathfrak {q}}_{1},\ldots ,{\mathfrak {q}}_{k}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e485248fa43aa065c9fb7cac39d8fe601868fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.498ex; height:3.009ex;" alt="{\displaystyle S_{\mathfrak {p}}=\{{\mathfrak {q}}_{1},\ldots ,{\mathfrak {q}}_{k}\}}">. This is called the <a href="/wiki/Splitting_of_prime_ideals_in_Galois_extensions" title="Splitting of prime ideals in Galois extensions">Splitting of prime ideals in Galois extensions</a>. <div class="mw-heading mw-heading4"><h4 id="Remarks_2">Remarks</h4>[<a href="/w/index.php?title=Integral_element&action=edit&section=28" title="Edit section: Remarks">edit</a>]</div> The same idea in the proof shows that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0381945929156997b99bb43e5b7067d18c9a84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.811ex; height:2.843ex;" alt="{\displaystyle L/K}"> is a purely inseparable extension (need not be normal), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e33d3fb1c68e0024bec7edf94421bb48ac78fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.195ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> is <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a>. Let A, K, etc. as before but assume L is only a finite field extension of K. Then <dl><dd>(i) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27e33d3fb1c68e0024bec7edf94421bb48ac78fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.195ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} B\to \operatorname {Spec} A}"> has finite fibers.</dd> <dd>(ii) the going-down holds between A and B: given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{1}\subset \cdots \subset {\mathfrak {p}}_{n}={\mathfrak {p}}'_{n}\cap A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> <mo>∩</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{1}\subset \cdots \subset {\mathfrak {p}}_{n}={\mathfrak {p}}'_{n}\cap A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cda6b18acb55f49f05e1423a19c1f25f386ca29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.323ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}_{1}\subset \cdots \subset {\mathfrak {p}}_{n}={\mathfrak {p}}'_{n}\cap A}">, there exists <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}'_{1}\subset \cdots \subset {\mathfrak {p}}'_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mo>′</mo> </msubsup> <mo>⊂</mo> <mo>⋯</mo> <mo>⊂</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}'_{1}\subset \cdots \subset {\mathfrak {p}}'_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aec9b73c9c499acfa77a96dd083b88df90e3091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.518ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}'_{1}\subset \cdots \subset {\mathfrak {p}}'_{n}}"> that contracts to it.</dd></dl> Indeed, in both statements, by enlarging L, we can assume L is a normal extension. Then (i) is immediate. As for (ii), by the going-up, we can find a chain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}''_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>″</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}''_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b223d8faac396a8a3cb16193d85960e49f3183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.3ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}''_{i}}"> that contracts to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}'_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}'_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3ff0e0f2e93e35a03ce556a59b0a0aca5ed7c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.962ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}'_{i}}">. By transitivity, there is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \in G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2ea6a3648b2b75cc8103c569a9630c194dcce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.997ex; height:2.176ex;" alt="{\displaystyle \sigma \in G}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ({\mathfrak {p}}''_{n})={\mathfrak {p}}'_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>″</mo> </msubsup> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ({\mathfrak {p}}''_{n})={\mathfrak {p}}'_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14b5abde36d1bf45330a5903f87e557e82d453a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.999ex; height:2.843ex;" alt="{\displaystyle \sigma ({\mathfrak {p}}''_{n})={\mathfrak {p}}'_{n}}"> and then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}'_{i}=\sigma ({\mathfrak {p}}''_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mo>″</mo> </msubsup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}'_{i}=\sigma ({\mathfrak {p}}''_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8500d1171e99bd2b2b1503b03c0a2187d1bb8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.499ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {p}}'_{i}=\sigma ({\mathfrak {p}}''_{i})}"> are the desired chain. <div class="mw-heading mw-heading2"><h2 id="Integral_closure">Integral closure</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=29" title="Edit section: Integral closure">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Integral_closure_of_an_ideal" title="Integral closure of an ideal">Integral closure of an ideal</a></div> Let A ⊂ B be rings and A' the integral closure of A in B. (See above for the definition.) Integral closures behave nicely under various constructions. Specifically, for a <a href="/wiki/Multiplicatively_closed_subset" class="mw-redirect" title="Multiplicatively closed subset">multiplicatively closed subset</a> S of A, the <a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization</a> S−1A' is the integral closure of S−1A in S−1B, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a3d65374e4ea41d77bc75e7f6b99e2f3663cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.561ex; height:3.009ex;" alt="{\displaystyle A'[t]}"> is the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13daf73f25e64310e7e5d67c5aa8dd4f97cb884f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle A[t]}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee02ed94f1960d266fc5a97534bf14cb3d99b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.897ex; height:2.843ex;" alt="{\displaystyle B[t]}">.<a href="#cite_note-14">[14]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"> are subrings of rings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i},1\leq i\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i},1\leq i\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adbef5716282e3aa99bbf8cc468fcc78ebbae7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.154ex; height:2.509ex;" alt="{\displaystyle B_{i},1\leq i\leq n}">, then the integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∏</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod A_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fea240e68b6764dd0ca2656179095c680a4c16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.899ex; height:3.843ex;" alt="{\displaystyle \prod A_{i}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∏</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod B_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071181755fe2a8030c007abdd71df3e3c7030f7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:5.92ex; height:3.843ex;" alt="{\displaystyle \prod B_{i}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod {A_{i}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∏</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod {A_{i}}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae18bf88e717797bb1a345babffa658823399dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.584ex; height:3.843ex;" alt="{\displaystyle \prod {A_{i}}'}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {A_{i}}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {A_{i}}'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86c6748e3a50c04161b0c171741bc501a319d1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.228ex; height:2.843ex;" alt="{\displaystyle {A_{i}}'}"> are the integral closures of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82cda0578ec6b48774c541ecb9bee4a90176e62f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.564ex; height:2.509ex;" alt="{\displaystyle B_{i}}">.<a href="#cite_note-15">[15]</a> The integral closure of a <a href="/wiki/Local_ring" title="Local ring">local ring</a> A in, say, B, need not be local. (If this is the case, the ring is called <a href="/wiki/Unibranch_local_ring" title="Unibranch local ring">unibranch</a>.) This is the case for example when A is <a href="/wiki/Henselian_ring" title="Henselian ring">Henselian</a> and B is a field extension of the field of fractions of A. If A is a subring of a field K, then the integral closure of A in K is the intersection of all <a href="/wiki/Valuation_ring" title="Valuation ring">valuation rings</a> of K containing A. Let A be an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }">-graded subring of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }">-<a href="/wiki/Graded_ring" title="Graded ring">graded ring</a> B. Then the integral closure of A in B is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }">-graded subring of B.<a href="#cite_note-16">[16]</a> There is also a concept of the <a href="/wiki/Integral_closure_of_an_ideal" title="Integral closure of an ideal">integral closure of an ideal</a>. The integral closure of an ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\subset R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf82feb3f84a530f44b000a8e003e00170e10ed5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle I\subset R}">, usually denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b8e66ae8bc148a0ef25e32292c223ee9ae24d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.383ex; height:3.009ex;" alt="{\displaystyle {\overline {I}}}">, is the set of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}"> such that there exists a monic polynomial <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x^{1}+a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x^{1}+a_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58849f4239dccdff1ed2a08169281da9016f8622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.947ex; height:3.009ex;" alt="{\displaystyle x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x^{1}+a_{n}}"></dd></dl> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}\in I^{i}}"> <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> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}\in I^{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33039a77c1451d7f6b8e701899eba38896239f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.886ex; height:3.009ex;" alt="{\displaystyle a_{i}\in I^{i}}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> as a root.<a href="#cite_note-17">[17]</a><a href="#cite_note-18">[18]</a> The <a href="/wiki/Radical_of_an_ideal" title="Radical of an ideal">radical of an ideal</a> is integrally closed.<a href="#cite_note-19">[19]</a><a href="#cite_note-20">[20]</a> For noetherian rings, there are alternate definitions as well. <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in {\overline {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in {\overline {I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852c4e9103b89b5d0e7e1e5d1f9f98f7a719df4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:3.009ex;" alt="{\displaystyle r\in {\overline {I}}}"> if there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/480fab09d4a38ea5b5c7555a182fd85fcdf9ce55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.612ex; height:2.176ex;" alt="{\displaystyle c\in R}"> not contained in any minimal prime, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cr^{n}\in I^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∈</mo> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cr^{n}\in I^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c79e963d69727703fe9f4c71a10d2678a606b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.55ex; height:2.343ex;" alt="{\displaystyle cr^{n}\in I^{n}}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}">.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in {\overline {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in {\overline {I}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852c4e9103b89b5d0e7e1e5d1f9f98f7a719df4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.273ex; height:3.009ex;" alt="{\displaystyle r\in {\overline {I}}}"> if in the normalized blow-up of I, the pull back of r is contained in the inverse image of I. The blow-up of an ideal is an operation of schemes which replaces the given ideal with a principal ideal. The normalization of a scheme is simply the scheme corresponding to the integral closure of all of its rings.</li></ul> The notion of integral closure of an ideal is used in some proofs of the <a href="/wiki/Going_up_and_going_down" title="Going up and going down">going-down theorem</a>. <div class="mw-heading mw-heading2"><h2 id="Conductor">Conductor</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=30" title="Edit section: Conductor">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Conductor_(ring_theory)" title="Conductor (ring theory)">Conductor (ring theory)</a></div> Let B be a ring and A a subring of B such that B is integral over A. Then the <a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a> of the A-module B/A is called the conductor of A in B. Because the notion has origin in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, the conductor is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}={\mathfrak {f}}(B/A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}={\mathfrak {f}}(B/A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2426fd02902725883d2a34a32f902d06aecbb153" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.094ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {f}}={\mathfrak {f}}(B/A)}">. Explicitly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7c8d65d0a0dbdbcd9c31e1791351016f30a445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.758ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {f}}}"> consists of elements a in A such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aB\subset A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>B</mi> <mo>⊂</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aB\subset A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df33de911f1d97a35bad6d3d1c4147c45294356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.835ex; height:2.176ex;" alt="{\displaystyle aB\subset A}">. (cf. <a href="/wiki/Idealizer" title="Idealizer">idealizer</a> in abstract algebra.) It is the largest <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of A that is also an ideal of B.<a href="#cite_note-21">[21]</a> If S is a multiplicatively closed subset of A, then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}{\mathfrak {f}}(B/A)={\mathfrak {f}}(S^{-1}B/S^{-1}A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{-1}{\mathfrak {f}}(B/A)={\mathfrak {f}}(S^{-1}B/S^{-1}A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a95ef36b3ff516b386dd33abdd12cf3b58e9f05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.136ex; height:3.176ex;" alt="{\displaystyle S^{-1}{\mathfrak {f}}(B/A)={\mathfrak {f}}(S^{-1}B/S^{-1}A)}">.</dd></dl> If B is a subring of the <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">total ring of fractions</a> of A, then we may identify <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}(B/A)=\operatorname {Hom} _{A}(B,A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>Hom</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}(B/A)=\operatorname {Hom} _{A}(B,A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df587ee5c722539c9976f1263e4e5dda0a2b963c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.992ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {f}}(B/A)=\operatorname {Hom} _{A}(B,A)}">.</dd></dl> Example: Let k be a field and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=k[t^{2},t^{3}]\subset B=k[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">]</mo> <mo>⊂</mo> <mi>B</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=k[t^{2},t^{3}]\subset B=k[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/165d2b60c645207d4f6b986a6a5a39e4cebd1ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.474ex; height:3.176ex;" alt="{\displaystyle A=k[t^{2},t^{3}]\subset B=k[t]}"> (i.e., A is the <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> of the <a href="/wiki/Affine_curve" class="mw-redirect" title="Affine curve">affine curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}=y^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}=y^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3da54c25f8beb3bd5dca4f4c4e3ba33c5145032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.697ex; height:3.009ex;" alt="{\displaystyle x^{2}=y^{3}}">). B is the integral closure of A in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af82cc40c9f5016723dec48de1d786702bfc2c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.86ex; height:2.843ex;" alt="{\displaystyle k(t)}">. The conductor of A in B is the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{2},t^{3})A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t^{2},t^{3})A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46627129228cbd24ca4b09cf9924b338334fb364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.374ex; height:3.176ex;" alt="{\displaystyle (t^{2},t^{3})A}">. More generally, the conductor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=k[[t^{a},t^{b}]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msup> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=k[[t^{a},t^{b}]]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f117f150e731c3272534bb1586093d810f1643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.393ex; height:3.176ex;" alt="{\displaystyle A=k[[t^{a},t^{b}]]}">, a, b relatively prime, is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t^{c},t^{c+1},\dots )A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t^{c},t^{c+1},\dots )A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48f3d066bff5ff3c7db3cfbecf8627b490560cfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.012ex; height:3.176ex;" alt="{\displaystyle (t^{c},t^{c+1},\dots )A}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=(a-1)(b-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=(a-1)(b-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed62b0004d43079504ac60284bce5cb0c034be6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.957ex; height:2.843ex;" alt="{\displaystyle c=(a-1)(b-1)}">.<a href="#cite_note-22">[22]</a> Suppose B is the integral closure of an integral domain A in the field of fractions of A such that the A-module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B/A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abec9996af8318eb6ecc5a41b2331d70fce8d92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.67ex; height:2.843ex;" alt="{\displaystyle B/A}"> is finitely generated. Then the conductor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7c8d65d0a0dbdbcd9c31e1791351016f30a445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.758ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {f}}}"> of A is an ideal defining the <a href="/wiki/Support_of_a_module" title="Support of a module">support of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B/A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B/A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abec9996af8318eb6ecc5a41b2331d70fce8d92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.67ex; height:2.843ex;" alt="{\displaystyle B/A}">; thus, A coincides with B in the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\mathfrak {f}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V({\mathfrak {f}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce1b051fe7cef138f03a1644ada31f48c2e9ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.355ex; height:2.843ex;" alt="{\displaystyle V({\mathfrak {f}})}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad53059b6a9d6c309ab164aa4eaed021844b5ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.78ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} A}">. In particular, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\mathfrak {p}}\in \operatorname {Spec} A\mid A_{\mathfrak {p}}{\text{ is integrally closed}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>∈</mo> <mi>Spec</mi> <mo>⁡</mo> <mi>A</mi> <mo>∣</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext> is integrally closed</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\mathfrak {p}}\in \operatorname {Spec} A\mid A_{\mathfrak {p}}{\text{ is integrally closed}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8de337546a8df07fd1f69f9644e12065b9fb4c1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.865ex; height:3.009ex;" alt="{\displaystyle \{{\mathfrak {p}}\in \operatorname {Spec} A\mid A_{\mathfrak {p}}{\text{ is integrally closed}}\}}">, the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V({\mathfrak {f}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">f</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V({\mathfrak {f}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cce1b051fe7cef138f03a1644ada31f48c2e9ee3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.355ex; height:2.843ex;" alt="{\displaystyle V({\mathfrak {f}})}">, is an <a href="/wiki/Open_set" title="Open set">open set</a>. <div class="mw-heading mw-heading2"><h2 id="Finiteness_of_integral_closure">Finiteness of integral closure</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=31" title="Edit section: Finiteness of integral closure">edit</a>]</div> An important but difficult question is on the finiteness of the integral closure of a finitely generated algebra. There are several known results. The integral closure of a <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domain</a> in a finite extension of the field of fractions is a Dedekind domain; in particular, a noetherian ring. This is a consequence of the <a href="/wiki/Krull%E2%80%93Akizuki_theorem" title="Krull–Akizuki theorem">Krull–Akizuki theorem</a>. In general, the integral closure of a noetherian domain of dimension at most 2 is noetherian; Nagata gave an example of dimension 3 noetherian domain whose integral closure is not noetherian.<a href="#cite_note-23">[23]</a> A nicer statement is this: the integral closure of a noetherian domain is a <a href="/wiki/Krull_domain" class="mw-redirect" title="Krull domain">Krull domain</a> (<a href="/wiki/Mori%E2%80%93Nagata_theorem" title="Mori–Nagata theorem">Mori–Nagata theorem</a>). Nagata also gave an example of dimension 1 noetherian local domain such that the integral closure is not finite over that domain.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] Let A be a noetherian integrally closed domain with field of fractions K. If L/K is a finite separable extension, then the integral closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"> of A in L is a finitely generated A-module.<a href="#cite_note-24">[24]</a> This is easy and standard (uses the fact that the trace defines a non-degenerate bilinear form). Let A be a finitely generated algebra over a field k that is an integral domain with field of fractions K. If L is a finite extension of K, then the integral closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"> of A in L is a finitely generated A-module and is also a finitely generated k-algebra.<a href="#cite_note-25">[25]</a> The result is due to Noether and can be shown using the <a href="/wiki/Noether_normalization_lemma" title="Noether normalization lemma">Noether normalization lemma</a> as follows. It is clear that it is enough to show the assertion when L/K is either separable or purely inseparable. The separable case is noted above, so assume L/K is purely inseparable. By the normalization lemma, A is integral over the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=k[x_{1},...,x_{d}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k[x_{1},...,x_{d}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d29efc83883fed4078458a5c8042df739c505f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.078ex; height:2.843ex;" alt="{\displaystyle S=k[x_{1},...,x_{d}]}">. Since L/K is a finite purely inseparable extension, there is a power q of a <a href="/wiki/Prime_number" title="Prime number">prime number</a> such that every element of L is a q-th root of an element in K. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51a2b832ed184c7b6481b3926bf8172d353fa7de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.896ex; height:2.509ex;" alt="{\displaystyle k'}"> be a finite extension of k containing all q-th roots of coefficients of finitely many rational functions that generate L. Then we have: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\subset k'(x_{1}^{1/q},...,x_{d}^{1/q}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>⊂</mo> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\subset k'(x_{1}^{1/q},...,x_{d}^{1/q}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d20c7093308b41b5c00c02d98f551ba8c460faaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.127ex; height:3.676ex;" alt="{\displaystyle L\subset k'(x_{1}^{1/q},...,x_{d}^{1/q}).}"> The ring on the right is the field of fractions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k'[x_{1}^{1/q},...,x_{d}^{1/q}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>k</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msubsup> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mrow> </msubsup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k'[x_{1}^{1/q},...,x_{d}^{1/q}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c64d0dd994c3e428567d402d45137e20923b3cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.284ex; height:3.676ex;" alt="{\displaystyle k'[x_{1}^{1/q},...,x_{d}^{1/q}]}">, which is the integral closure of S; thus, contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}">. Hence, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"> is finite over S; a fortiori, over A. The result remains true if we replace k by Z. The integral closure of a complete local noetherian domain A in a finite extension of the field of fractions of A is finite over A.<a href="#cite_note-26">[26]</a> More precisely, for a local noetherian ring R, we have the following chains of implications:<a href="#cite_note-27">[27]</a> <dl><dd>(i) A complete <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"> A is a <a href="/wiki/Nagata_ring" title="Nagata ring">Nagata ring</a></dd> <dd>(ii) A is a Nagata domain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"> A <a href="/wiki/Analytically_unramified" class="mw-redirect" title="Analytically unramified">analytically unramified</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"> the integral closure of the completion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5575024af93e0125d130908ca42fc3c347b800a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.019ex; width:1.859ex; height:3.009ex;" alt="{\displaystyle {\widehat {A}}}"> is finite over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5575024af93e0125d130908ca42fc3c347b800a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.019ex; width:1.859ex; height:3.009ex;" alt="{\displaystyle {\widehat {A}}}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"> the integral closure of A is finite over A.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Noether's_normalization_lemma">Noether's normalization lemma</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=32" title="Edit section: Noether's normalization lemma">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Noether_normalization_lemma" title="Noether normalization lemma">Noether normalization lemma</a></div> Noether's normalisation lemma is a theorem in <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>. Given a field K and a finitely generated K-algebra A, the theorem says it is possible to find elements y1, y2, ..., ym in A that are <a href="/wiki/Algebraic_independence" title="Algebraic independence">algebraically independent</a> over K such that A is finite (and hence integral) over B = K[y1,..., ym]. Thus the extension K ⊂ A can be written as a composite K ⊂ B ⊂ A where K ⊂ B is a purely transcendental extension and B ⊂ A is finite.<a href="#cite_note-28">[28]</a> <div class="mw-heading mw-heading2"><h2 id="Integral_morphisms">Integral morphisms</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=33" title="Edit section: Integral morphisms">edit</a>]</div> In <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}"> of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> is integral if it is <a href="/wiki/Sheaf_of_algebras#Affine_morphism" title="Sheaf of algebras">affine</a> and if for some (equivalently, every) affine open cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.387ex; height:2.509ex;" alt="{\displaystyle U_{i}}"> of Y, every map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(U_{i})\to U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(U_{i})\to U_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/093a1ff6dbb9f9722a5293a2a95b58a31d8b1a2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.851ex; height:3.176ex;" alt="{\displaystyle f^{-1}(U_{i})\to U_{i}}"> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} (A)\to \operatorname {Spec} (B)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Spec</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} (A)\to \operatorname {Spec} (B)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397faa9ee0dc72e2e18272dd5ce03aba3ac99700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.039ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spec} (A)\to \operatorname {Spec} (B)}"> where A is an integral B-algebra. The class of integral morphisms is more general than the class of <a href="/wiki/Finite_morphism" title="Finite morphism">finite morphisms</a> because there are integral extensions that are not finite, such as, in many cases, the algebraic closure of a field over the field. <div class="mw-heading mw-heading2"><h2 id="Absolute_integral_closure">Absolute integral closure</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=34" title="Edit section: Absolute integral closure">edit</a>]</div> Let A be an integral domain and L (some) <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of the field of fractions of A. Then the integral closure <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.254ex; height:2.509ex;" alt="{\displaystyle A^{+}}"> of A in L is called the absolute integral closure of A.<a href="#cite_note-29">[29]</a> It is unique up to a non-canonical <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>. The <a href="/wiki/Ring_of_all_algebraic_integers" class="mw-redirect" title="Ring of all algebraic integers">ring of all algebraic integers</a> is an example (and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b380a5ff4e2d7d22a0dc1aea46e7ecba61f95fe6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.254ex; height:2.509ex;" alt="{\displaystyle A^{+}}"> is typically not noetherian). <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=35" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Normal_scheme" title="Normal scheme">Normal scheme</a></li> <li><a href="/wiki/Noether_normalization_lemma" title="Noether normalization lemma">Noether normalization lemma</a></li> <li><a href="/wiki/Algebraic_integer" title="Algebraic integer">Algebraic integer</a></li> <li><a href="/wiki/Splitting_of_prime_ideals_in_Galois_extensions" title="Splitting of prime ideals in Galois extensions">Splitting of prime ideals in Galois extensions</a></li> <li><a href="/wiki/Torsor_(algebraic_geometry)" title="Torsor (algebraic geometry)">Torsor (algebraic geometry)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=36" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> The above equation is sometimes called an integral equation and b is said to be integrally dependent on A (as opposed to <a href="/wiki/Algebraic_dependent" class="mw-redirect" title="Algebraic dependent">algebraic dependent</a>). </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFMilne2020">Milne 2020</a>, Theorem 6.4 </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFKaplansky1974">Kaplansky 1974</a>, 1.2. Exercise 4. </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHartshorne1977">Hartshorne 1977</a>, Ch. II, Exercise 5.14 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> This proof is due to Dedekind (Milne, ANT). Alternatively, one can use symmetric polynomials to show integral elements form a ring. (loc cit.) </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> Chapter 2 of <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFKaplansky1974">Kaplansky 1974</a>, Theorem 42 </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFBourbaki2006">Bourbaki 2006</a>, Ch 5, §2, Corollary 4 to Theorem 1. </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFMatsumura1970">Matsumura 1970</a>, Ch 2. Theorem 7 </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFBourbaki2006">Bourbaki 2006</a>, Ch 5, §1, Proposition 5 </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFAtiyahMacdonald1994">Atiyah & Macdonald 1994</a>, Ch 5. Exercise 35 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/05JW">"Section 32.14 (05JW): Universally closed morphisms—The Stacks project"</a>. stacks.math.columbia.edu. Retrieved 2020-05-11.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=stacks.math.columbia.edu&rft.atitle=Section+32.14+%2805JW%29%3A+Universally+closed+morphisms%E2%80%94The+Stacks+project&rft_id=https%3A%2F%2Fstacks.math.columbia.edu%2Ftag%2F05JW&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStein" class="citation book cs1">Stein. <a rel="nofollow" class="external text" href="https://wstein.org/books/ant/ant.pdf">Computational Introduction to Algebraic Number Theory</a> (PDF). p. 101.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Introduction+to+Algebraic+Number+Theory&rft.pages=101&rft.au=Stein&rft_id=https%3A%2F%2Fwstein.org%2Fbooks%2Fant%2Fant.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> An exercise in <a href="#CITEREFAtiyahMacdonald1994">Atiyah & Macdonald 1994</a> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFBourbaki2006">Bourbaki 2006</a>, Ch 5, §1, Proposition 9 </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> Proof: Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :B\to B[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>B</mi> <mo stretchy="false">→</mo> <mi>B</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :B\to B[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e623bb398cd6851377fbf06b271d0cb4be6e2f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.598ex; height:2.843ex;" alt="{\displaystyle \phi :B\to B[t]}"> be a ring homomorphism such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (b_{n})=b_{n}t^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (b_{n})=b_{n}t^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39b101807d904bfdf40484c4f3c845097b784d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.783ex; height:2.843ex;" alt="{\displaystyle \phi (b_{n})=b_{n}t^{n}}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"> is homogeneous of degree n. The integral closure of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13daf73f25e64310e7e5d67c5aa8dd4f97cb884f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle A[t]}"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ee02ed94f1960d266fc5a97534bf14cb3d99b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.897ex; height:2.843ex;" alt="{\displaystyle B[t]}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a3d65374e4ea41d77bc75e7f6b99e2f3663cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.561ex; height:3.009ex;" alt="{\displaystyle A'[t]}">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a12527148d6ed68adc91d9b419eb4b92d58ef6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.509ex;" alt="{\displaystyle A'}"> is the integral closure of A in B. If b in B is integral over A, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591cd82f64428399874ae7c2cec10b7f23dc53df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.192ex; height:2.843ex;" alt="{\displaystyle \phi (b)}"> is integral over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13daf73f25e64310e7e5d67c5aa8dd4f97cb884f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.876ex; height:2.843ex;" alt="{\displaystyle A[t]}">; i.e., it is in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A'[t]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A'[t]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a3d65374e4ea41d77bc75e7f6b99e2f3663cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.561ex; height:3.009ex;" alt="{\displaystyle A'[t]}">. That is, each coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e2d72f6dd9375c8f1f59f1effd9b4e5492ac97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.509ex;" alt="{\displaystyle b_{n}}"> in the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi (b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/591cd82f64428399874ae7c2cec10b7f23dc53df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.192ex; height:2.843ex;" alt="{\displaystyle \phi (b)}"> is in A. </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> Exercise 4.14 in <a href="#CITEREFEisenbud1995">Eisenbud 1995</a> </li> <li id="cite_note-18"><a href="#cite_ref-18">^</a> Definition 1.1.1 in <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> Exercise 4.15 in <a href="#CITEREFEisenbud1995">Eisenbud 1995</a> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> Remark 1.1.3 in <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a> </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> Chapter 12 of <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a>, Example 12.2.1 </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a>, Exercise 4.9 </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> <a href="#CITEREFAtiyahMacdonald1994">Atiyah & Macdonald 1994</a>, Ch 5. Proposition 5.17 </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <a href="#CITEREFHartshorne1977">Hartshorne 1977</a>, Ch I. Theorem 3.9 A </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <a href="#CITEREFHunekeSwanson2006">Huneke & Swanson 2006</a>, Theorem 4.3.4 </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFMatsumura1970">Matsumura 1970</a>, Ch 12 </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> Chapter 4 of Reid. </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <a href="/wiki/Melvin_Hochster" title="Melvin Hochster">Melvin Hochster</a>, <a rel="nofollow" class="external text" href="http://www.math.lsa.umich.edu/~hochster/711F07/L09.07.pdf">Math 711: Lecture of September 7, 2007</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=37" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtiyahMacdonald1994" class="citation book cs1"><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael Francis</a>; <a href="/wiki/Ian_G._Macdonald" title="Ian G. Macdonald">Macdonald, Ian G.</a> (1994) [1969]. Introduction to commutative algebra. Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-40751-5" title="Special:BookSources/0-201-40751-5"><bdi>0-201-40751-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+commutative+algebra&rft.pub=Addison%E2%80%93Wesley&rft.date=1994&rft.isbn=0-201-40751-5&rft.aulast=Atiyah&rft.aufirst=Michael+Francis&rft.au=Macdonald%2C+Ian+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki2006" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (2006). Algèbre commutative. Berlin: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-33937-3" title="Special:BookSources/978-3-540-33937-3"><bdi>978-3-540-33937-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre+commutative&rft.place=Berlin&rft.pub=Springer&rft.date=2006&rft.isbn=978-3-540-33937-3&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisenbud1995" class="citation cs2">Eisenbud, David (1995), Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94268-8" title="Special:BookSources/0-387-94268-8"><bdi>0-387-94268-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+Algebra+with+a+View+Toward+Algebraic+Geometry&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=0-387-94268-8&rft.aulast=Eisenbud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaplansky1974" class="citation book cs1">Kaplansky, Irving (September 1974). <a rel="nofollow" class="external text" href="https://archive.org/details/commutativerings00irvi">Commutative Rings</a>. Lectures in Mathematics. <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-42454-5" title="Special:BookSources/0-226-42454-5"><bdi>0-226-42454-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+Rings&rft.series=Lectures+in+Mathematics&rft.pub=University+of+Chicago+Press&rft.date=1974-09&rft.isbn=0-226-42454-5&rft.aulast=Kaplansky&rft.aufirst=Irving&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcommutativerings00irvi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90244-9" title="Special:BookSources/978-0-387-90244-9"><bdi>978-0-387-90244-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-0-387-90244-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0463157%23id-name%3DMR&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatsumura1970" class="citation cs2">Matsumura, H (1970), Commutative algebra</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+algebra&rft.date=1970&rft.aulast=Matsumura&rft.aufirst=H&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li>H. Matsumura Commutative ring theory. Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilne2020" class="citation web cs1"><a href="/wiki/James_Milne_(mathematician)" title="James Milne (mathematician)">Milne, J. S.</a> (19 July 2020). <a rel="nofollow" class="external text" href="https://www.jmilne.org/math/CourseNotes/ant.html">"Algebraic number theory"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Algebraic+number+theory&rft.date=2020-07-19&rft.aulast=Milne&rft.aufirst=J.+S.&rft_id=https%3A%2F%2Fwww.jmilne.org%2Fmath%2FCourseNotes%2Fant.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHunekeSwanson2006" class="citation cs2">Huneke, Craig; <a href="/wiki/Irena_Swanson" title="Irena Swanson">Swanson, Irena</a> (2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191115053353/http://people.reed.edu/~iswanson/book/index.html">Integral closure of ideals, rings, and modules</a>, London Mathematical Society Lecture Note Series, vol. 336, Cambridge, UK: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-68860-4" title="Special:BookSources/978-0-521-68860-4"><bdi>978-0-521-68860-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2266432">2266432</a>, archived from <a rel="nofollow" class="external text" href="http://people.reed.edu/~iswanson/book/index.html">the original</a> on 2019-11-15, retrieved 2011-03-01</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integral+closure+of+ideals%2C+rings%2C+and+modules&rft.place=Cambridge%2C+UK&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-68860-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2266432%23id-name%3DMR&rft.aulast=Huneke&rft.aufirst=Craig&rft.au=Swanson%2C+Irena&rft_id=http%3A%2F%2Fpeople.reed.edu%2F~iswanson%2Fbook%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIntegral+element" class="Z3988"></li> <li><a href="/wiki/Miles_Reid" title="Miles Reid">M. Reid</a>, Undergraduate Commutative Algebra, London Mathematical Society, 29, Cambridge University Press, 1995.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Integral_element&action=edit&section=38" title="Edit section: Further reading">edit</a>]</div> <ul><li>Irena Swanson, <a rel="nofollow" class="external text" href="http://people.reed.edu/~iswanson/trieste.pdf">Integral closures of ideals and rings</a></li> <li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/7775">Do DG-algebras have any sensible notion of integral closure?</a></li> <li><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/66445">Is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{1},\ldots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[x_{1},\ldots ,x_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e758c6a747cc0db80779557345d9b98f4f62f3ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.968ex; height:2.843ex;" alt="{\displaystyle k[x_{1},\ldots ,x_{n}}"></a> always an integral extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[f_{1},\ldots ,f_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[f_{1},\ldots ,f_{n}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9043dd5854c76907bdb632eea2ffa2d910cb73a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.234ex; height:2.843ex;" alt="{\displaystyle k[f_{1},\ldots ,f_{n}]}"> for a regular sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{1},\ldots ,f_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{1},\ldots ,f_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f659d1e48dc53e3e5c92a3812c3ce761f719ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.539ex; height:2.843ex;" alt="{\displaystyle (f_{1},\ldots ,f_{n})}">?]</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; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Integral_element&oldid=1260124087">https://en.wikipedia.org/w/index.php?title=Integral_element&oldid=1260124087</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Commutative_algebra" title="Category:Commutative algebra">Commutative algebra</a></li><li><a href="/wiki/Category:Ring_theory" title="Category:Ring theory">Ring theory</a></li><li><a href="/wiki/Category:Algebraic_structures" title="Category:Algebraic structures">Algebraic structures</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_October_2012" title="Category:Articles with unsourced statements from October 2012">Articles with unsourced statements from October 2012</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_August_2013" title="Category:Articles with unsourced statements from August 2013">Articles with unsourced statements from August 2013</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 29 November 2024, at 00:33 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; 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