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<span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Uniqueness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Uniqueness</span> </div> </a> <ul id="toc-Uniqueness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Irreducibility" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Irreducibility"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Irreducibility</span> </div> </a> <ul id="toc-Irreducibility-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Minimal_polynomial_generates_Jα" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimal_polynomial_generates_Jα"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Minimal polynomial generates <i>J</i><sub><i>α</i></sub></span> </div> </a> <ul id="toc-Minimal_polynomial_generates_Jα-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Minimal_polynomial_of_a_Galois_field_extension" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Minimal_polynomial_of_a_Galois_field_extension"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Minimal polynomial of a Galois field extension</span> </div> </a> <ul id="toc-Minimal_polynomial_of_a_Galois_field_extension-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quadratic_field_extensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quadratic_field_extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Quadratic field extensions</span> </div> </a> <ul id="toc-Quadratic_field_extensions-sublist" class="vector-toc-list"> <li id="toc-Q(√2)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Q(√2)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Q(<span>√<span>2</span></span>)</span> </div> </a> <ul id="toc-Q(√2)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Q(√d_)" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Q(√d_)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Q(<span>√<span><i>d</i></span></span> )</span> </div> </a> <ul id="toc-Q(√d_)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Biquadratic_field_extensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biquadratic_field_extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Biquadratic field extensions</span> </div> </a> <ul id="toc-Biquadratic_field_extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Roots_of_unity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Roots_of_unity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Roots of unity</span> </div> </a> <ul id="toc-Roots_of_unity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Swinnerton-Dyer_polynomials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Swinnerton-Dyer_polynomials"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Swinnerton-Dyer polynomials</span> </div> </a> <ul id="toc-Swinnerton-Dyer_polynomials-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul id="toc-References-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" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet 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data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Polinomi_minimal" title="Polinomi minimal – Catalan" lang="ca" hreflang="ca" data-title="Polinomi minimal" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Minim%C3%A1ln%C3%AD_polynom_(teorie_t%C4%9Bles)" title="Minimální polynom (teorie těles) – Czech" lang="cs" hreflang="cs" data-title="Minimální polynom (teorie těles)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Minimalpolynom" title="Minimalpolynom – German" lang="de" hreflang="de" data-title="Minimalpolynom" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CE%BB%CE%AC%CF%87%CE%B9%CF%83%CF%84%CE%BF_%CF%80%CE%BF%CE%BB%CF%85%CF%8E%CE%BD%CF%85%CE%BC%CE%BF" title="Ελάχιστο πολυώνυμο – Greek" lang="el" hreflang="el" data-title="Ελάχιστο πολυώνυμο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Polinomio_m%C3%ADnimo_(teor%C3%ADa_de_cuerpos)" title="Polinomio mínimo (teoría de cuerpos) – Spanish" lang="es" hreflang="es" data-title="Polinomio mínimo (teoría de cuerpos)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%AF%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C_%DA%A9%D9%85%DB%8C%D9%86%D9%87_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D9%85%DB%8C%D8%AF%D8%A7%D9%86)" title="چندجمله‌ای کمینه (نظریه میدان) – Persian" lang="fa" hreflang="fa" data-title="چندجمله‌ای کمینه (نظریه میدان)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me_minimal_(th%C3%A9orie_des_corps)" title="Polynôme minimal (théorie des corps) – French" lang="fr" hreflang="fr" data-title="Polynôme minimal (théorie des corps)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70893996 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EC%B5%9C%EC%86%8C_%EB%8B%A4%ED%95%AD%EC%8B%9D_(%EC%B2%B4%EB%A1%A0)" title="최소 다항식 (체론) – Korean" lang="ko" hreflang="ko" data-title="최소 다항식 (체론)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9D_%D7%9E%D7%99%D7%A0%D7%99%D7%9E%D7%9C%D7%99" title="פולינום מינימלי – Hebrew" lang="he" hreflang="he" data-title="פולינום מינימלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Minimale_polynoom_(galoistheorie)" title="Minimale polynoom (galoistheorie) – Dutch" lang="nl" hreflang="nl" data-title="Minimale polynoom 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minimal – Romanian" lang="ro" hreflang="ro" data-title="Polinom minimal" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B8%D0%BD%D0%B8%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD_%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B3%D0%BE_%D1%8D%D0%BB%D0%B5%D0%BC%D0%B5%D0%BD%D1%82%D0%B0" title="Минимальный многочлен алгебраического элемента – Russian" lang="ru" hreflang="ru" data-title="Минимальный многочлен алгебраического элемента" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D1%96%D0%BD%D1%96%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD_(%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%BF%D0%BE%D0%BB%D1%96%D0%B2)" title="Мінімальний многочлен (теорія полів) – Ukrainian" lang="uk" hreflang="uk" data-title="Мінімальний многочлен (теорія полів)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90a_th%E1%BB%A9c_t%E1%BB%91i_ti%E1%BB%83u_(l%C3%BD_thuy%E1%BA%BFt_tr%C6%B0%E1%BB%9Dng)" title="Đa thức tối tiểu (lý thuyết trường) – Vietnamese" lang="vi" hreflang="vi" data-title="Đa thức tối tiểu (lý thuyết trường)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh 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</div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">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"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Concept in abstract algebra</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">For the minimal polynomial of a matrix, see <a href="/wiki/Minimal_polynomial_(linear_algebra)" title="Minimal polynomial (linear algebra)">Minimal polynomial (linear algebra)</a>.</div> <p>In <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>minimal polynomial</b> of an element <span class="texhtml"><i>α</i></span> of an <a href="/wiki/Field_extension" title="Field extension">extension field</a> of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is, roughly speaking, the <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of lowest <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> having coefficients in the smaller field, such that <span class="texhtml"><i>α</i></span> is a root of the polynomial. If the minimal polynomial of <span class="texhtml"><i>α</i></span> exists, it is unique. The coefficient of the highest-degree term in the polynomial is required to be 1. </p><p>More formally, a minimal polynomial is defined relative to a <a href="/wiki/Field_extension" title="Field extension">field extension</a> <span class="texhtml"><i>E</i>/<i>F</i></span> and an element of the extension field <span class="texhtml"><i>E</i>/<i>F</i></span>. The minimal polynomial of an element, if it exists, is a member of <span class="texhtml"><i>F</i>[<i>x</i>]</span>, the <a href="/wiki/Polynomial_ring" title="Polynomial ring">ring of polynomials</a> in the variable <span class="texhtml"><i>x</i></span> with coefficients in <span class="texhtml"><i>F</i></span>. Given an element <span class="texhtml"><i>α</i></span> of <span class="texhtml"><i>E</i></span>, let <span class="texhtml"><i>J</i><sub><i>α</i></sub></span> be the set of all polynomials <span class="texhtml"><i>f</i>(<i>x</i>)</span> in <span class="texhtml"><i>F</i>[<i>x</i>]</span> such that <span class="texhtml"><i>f</i>(<i>α</i>) = 0</span>. The element <span class="texhtml"><i>α</i></span> is called a <a href="/wiki/Zero_of_a_function" title="Zero of a function">root or zero</a> of each polynomial in <span class="texhtml"><i>J</i><sub><i>α</i></sub></span> </p><p>More specifically, <i>J</i><sub><i>α</i></sub> is the kernel of the <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from <i>F</i>[<i>x</i>] to <i>E</i> which sends polynomials <i>g</i> to their value <i>g</i>(<i>α</i>) at the element <i>α</i>. Because it is the kernel of a ring homomorphism, <i>J</i><sub><i>α</i></sub> is an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of the polynomial ring <i>F</i>[<i>x</i>]: it is closed under polynomial addition and subtraction (hence containing the zero polynomial), as well as under multiplication by elements of <i>F</i> (which is <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> if <i>F</i>[<i>x</i>] is regarded as a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over <i>F</i>). </p><p>The zero polynomial, all of whose coefficients are 0, is in every <span class="texhtml"><i>J</i><sub><i>α</i></sub></span> since <span class="texhtml">0<i>α</i><sup><i>i</i></sup> = 0</span> for all <span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>i</i></span>. This makes the zero polynomial useless for classifying different values of <span class="texhtml"><i>α</i></span> into types, so it is excepted. If there are any non-zero polynomials in <span class="texhtml"><i>J</i><sub><i>α</i></sub></span>, i.e. if the latter is not the zero ideal, then <span class="texhtml"><i>α</i></span> is called an <a href="/wiki/Algebraic_element" title="Algebraic element">algebraic element</a> over <span class="texhtml"><i>F</i></span>, and there exists a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> of least degree in <span class="texhtml"><i>J</i><sub><i>α</i></sub></span>. This is the minimal polynomial of <span class="texhtml"><i>α</i></span> with respect to <span class="texhtml"><i>E</i>/<i>F</i></span>. It is unique and <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over <span class="texhtml"><i>F</i></span>. If the zero polynomial is the only member of <span class="texhtml"><i>J</i><sub><i>α</i></sub></span>, then <span class="texhtml"><i>α</i></span> is called a <a href="/wiki/Transcendental_element" class="mw-redirect" title="Transcendental element">transcendental element</a> over <span class="texhtml"><i>F</i></span> and has no minimal polynomial with respect to <span class="texhtml"><i>E</i>/<i>F</i></span>. </p><p>Minimal polynomials are useful for constructing and analyzing field extensions. When <span class="texhtml"><i>α</i></span> is algebraic with minimal polynomial <span class="texhtml"><i>f</i>(<i>x</i>)</span>, the smallest field that contains both <span class="texhtml"><i>F</i></span> and <span class="texhtml"><i>α</i></span> is <a href="/wiki/Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">isomorphic</a> to the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> <span class="texhtml"><i>F</i>[<i>x</i>]/⟨<i>f</i>(<i>x</i>)⟩</span>, where <span class="texhtml">⟨<i>f</i>(<i>x</i>)⟩</span> is the ideal of <span class="texhtml"><i>F</i>[<i>x</i>]</span> generated by <span class="texhtml"><i>f</i>(<i>x</i>)</span>. Minimal polynomials are also used to define <a href="/wiki/Conjugate_elements" class="mw-redirect" title="Conjugate elements">conjugate elements</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>E</i>/<i>F</i> be a <a href="/wiki/Field_extension" title="Field extension">field extension</a>, <i>α</i> an element of <i>E</i>, and <i>F</i>[<i>x</i>] the ring of polynomials in <i>x</i> over <i>F</i>. The element <i>α</i> has a minimal polynomial when <i>α</i> is algebraic over <i>F</i>, that is, when <i>f</i>(<i>α</i>) = 0 for some non-zero polynomial <i>f</i>(<i>x</i>) in <i>F</i>[<i>x</i>]. Then the minimal polynomial of <i>α</i> is defined as the monic polynomial of least degree among all polynomials in <i>F</i>[<i>x</i>] having <i>α</i> as a root. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=2" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Throughout this section, let <i>E</i>/<i>F</i> be a field extension over <i>F</i> as above, let <i>α</i> ∈ <i>E</i> be an algebraic element over <i>F</i> and let <i>J</i><sub><i>α</i></sub> be the ideal of polynomials vanishing on <i>α</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Uniqueness">Uniqueness</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=3" title="Edit section: Uniqueness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The minimal polynomial <i>f</i> of <i>α</i> is unique. </p><p>To prove this, suppose that <i>f</i> and <i>g</i> are monic polynomials in <i>J</i><sub><i>α</i></sub> of minimal degree <i>n</i> > 0. We have that <i>r</i> := <i>f</i>−<i>g</i> ∈ <i>J</i><sub><i>α</i></sub> (because the latter is closed under addition/subtraction) and that <i>m</i> := deg(<i>r</i>) < <i>n</i> (because the polynomials are monic of the same degree). If <i>r</i> is not zero, then <i>r</i> / <i>c</i><sub><i>m</i></sub> (writing <i>c</i><sub><i>m</i></sub> ∈ <i>F</i> for the non-zero coefficient of highest degree in <i>r</i>) is a monic polynomial of degree <i>m</i> < <i>n</i> such that <i>r</i> / <i>c</i><sub><i>m</i></sub> ∈ <i>J</i><sub><i>α</i></sub> (because the latter is closed under multiplication/division by non-zero elements of <i>F</i>), which contradicts our original assumption of minimality for <i>n</i>. We conclude that 0 = <i>r</i> = <i>f</i> − <i>g</i>, i.e. that <i>f</i> = <i>g</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Irreducibility">Irreducibility</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=4" title="Edit section: Irreducibility"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The minimal polynomial <i>f</i> of <i>α</i> is irreducible, i.e. it cannot be factorized as <i>f</i> = <i>gh</i> for two polynomials <i>g</i> and <i>h</i> of strictly lower degree. </p><p>To prove this, first observe that any factorization <i>f</i> = <i>gh</i> implies that either <i>g</i>(<i>α</i>) = 0 or <i>h</i>(<i>α</i>) = 0, because <i>f</i>(<i>α</i>) = 0 and <i>F</i> is a field (hence also an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>). Choosing both <i>g</i> and <i>h</i> to be of degree strictly lower than <i>f</i> would then contradict the minimality requirement on <i>f</i>, so <i>f</i> must be irreducible. </p> <div class="mw-heading mw-heading3"><h3 id="Minimal_polynomial_generates_Jα"><span id="Minimal_polynomial_generates_J.CE.B1"></span>Minimal polynomial generates <i>J</i><sub><i>α</i></sub></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=5" title="Edit section: Minimal polynomial generates Jα"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The minimal polynomial <i>f</i> of <i>α</i> generates the ideal <i>J</i><sub><i>α</i></sub>, i.e. every <i> g</i> in <i>J</i><sub><i>α</i></sub> can be factorized as <i>g=fh</i> for some <i>h' </i> in <i>F</i>[<i>x</i>]. </p><p>To prove this, it suffices to observe that <i>F</i>[<i>x</i>] is a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>, because <i>F</i> is a field: this means that every ideal <i>I</i> in <i>F</i>[<i>x</i>], <i>J</i><sub><i>α</i></sub> amongst them, is generated by a single element <i>f</i>. With the exception of the zero ideal <i>I</i> = {0}, the generator <i>f</i> must be non-zero and it must be the unique polynomial of minimal degree, up to a factor in <i>F</i> (because the degree of <i>fg</i> is strictly larger than that of <i>f</i> whenever <i>g</i> is of degree greater than zero). In particular, there is a unique monic generator <i>f</i>, and all generators must be irreducible. When <i>I</i> is chosen to be <i>J</i><sub><i>α</i></sub>, for <i>α</i> algebraic over <i>F</i>, then the monic generator <i>f</i> is the minimal polynomial of <i>α</i>. </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Minimal_polynomial_of_a_Galois_field_extension">Minimal polynomial of a Galois field extension</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=7" title="Edit section: Minimal polynomial of a Galois field extension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Given a Galois field extension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/K}"> <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></span><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}"></span> the minimal polynomial of any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbef4ff63e30cd41cf6c012293e4c258b31801c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.911ex; height:2.176ex;" alt="{\displaystyle \alpha \in L}"></span> not in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/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}"></span> can be computed as</p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\prod _{\sigma \in {\text{Gal}}(L/K)}(x-\sigma (\alpha ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gal</mtext> </mrow> <mo stretchy="false">(</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>K</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=\prod _{\sigma \in {\text{Gal}}(L/K)}(x-\sigma (\alpha ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d78c4d15bda8abdae2b33e88fc6b65b62de963f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.409ex; height:6.009ex;" alt="{\displaystyle f(x)=\prod _{\sigma \in {\text{Gal}}(L/K)}(x-\sigma (\alpha ))}"></span></p></blockquote><p>if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> has no stabilizers in the Galois action. Since it is irreducible, which can be deduced by looking at the roots of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/258eaada38956fb69b8cb1a2eef46bcb97d3126b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}"></span>, it is the minimal polynomial. Note that the same kind of formula can be found by replacing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G={\text{Gal}}(L/K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gal</mtext> </mrow> <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={\text{Gal}}(L/K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ea08958dcafb2ae8dae80211629b10e429ceef" 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={\text{Gal}}(L/K)}"></span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N={\text{Stab}}(\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>Stab</mtext> </mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N={\text{Stab}}(\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26da01f6e45797c0e38315e76689b2374d923b22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.111ex; height:2.843ex;" alt="{\displaystyle N={\text{Stab}}(\alpha )}"></span> is the stabilizer group of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. For example, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3581f6410248bed60440badc5bd362ca1cc8c82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.394ex; height:2.176ex;" alt="{\displaystyle \alpha \in K}"></span> then its stabilizer is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, hence <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7452ea0e1fdee917d5766a3dbb9ebbfe46f4590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.467ex; height:2.843ex;" alt="{\displaystyle (x-\alpha )}"></span> is its minimal polynomial. </p><div class="mw-heading mw-heading3"><h3 id="Quadratic_field_extensions">Quadratic field extensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=8" title="Edit section: Quadratic field extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Q(√2)"><span id="Q.28.E2.88.9A2.29"></span>Q(<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>)</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=9" title="Edit section: Q(√2)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>F</i> = <b>Q</b>, <i>E</i> = <b>R</b>, <i>α</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>, then the minimal polynomial for <i>α</i> is <i>a</i>(<i>x</i>) = <i>x</i><sup>2</sup> − 2. The base field <i>F</i> is important as it determines the possibilities for the coefficients of <i>a</i>(<i>x</i>). For instance, if we take <i>F</i> = <b>R</b>, then the minimal polynomial for <i>α</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> is <i>a</i>(<i>x</i>) = <i>x</i> − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Q(√d_)"><span id="Q.28.E2.88.9Ad_.29"></span>Q(<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>d</i></span></span> )</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=10" title="Edit section: Q(√d )"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div><p> In general, for the quadratic extension given by a square-free <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span>, computing the minimal polynomial of an element <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+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> <mspace width="thinmathspace" /> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b{\sqrt {d\,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb783a9c644d6bf739896387066513e2c6737a65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.607ex; height:3.009ex;" alt="{\displaystyle a+b{\sqrt {d\,}}}"></span> can be found using Galois theory. Then</p><blockquote><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f(x)&=(x-(a+b{\sqrt {d\,}}))(x-(a-b{\sqrt {d\,}}))\\&=x^{2}-2ax+(a^{2}-b^{2}d)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>d</mi> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&=(x-(a+b{\sqrt {d\,}}))(x-(a-b{\sqrt {d\,}}))\\&=x^{2}-2ax+(a^{2}-b^{2}d)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da742607f6eac9f44ac47c25e5ad6cdd24f7bc19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.399ex; margin-bottom: -0.272ex; width:41.058ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}f(x)&=(x-(a+b{\sqrt {d\,}}))(x-(a-b{\sqrt {d\,}}))\\&=x^{2}-2ax+(a^{2}-b^{2}d)\end{aligned}}}"></span></p></blockquote><p>in particular, this implies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306e8f27c80409c7c09faef47f906fbd2b071863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.783ex; height:2.176ex;" alt="{\displaystyle 2a\in \mathbb {Z} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}-b^{2}d\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}-b^{2}d\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f49eede4d8430917182898216d04aef6cdb85b2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.783ex; height:2.843ex;" alt="{\displaystyle a^{2}-b^{2}d\in \mathbb {Z} }"></span>. This can be used to determine <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d\,}}\!\!\!\;\;)}}"> <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> <mi>d</mi> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d\,}}\!\!\!\;\;)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7885456cd39aedd5d76889bde0c7065e286fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.385ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbb {Q} ({\sqrt {d\,}}\!\!\!\;\;)}}"></span> through a <a href="/wiki/Quadratic_integer#Determining_the_ring_of_integers" title="Quadratic integer">series of relations using modular arithmetic</a>. </p><div class="mw-heading mw-heading3"><h3 id="Biquadratic_field_extensions">Biquadratic field extensions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=11" title="Edit section: Biquadratic field extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>α</i> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>, then the minimal polynomial in <b>Q</b>[<i>x</i>] is <i>a</i>(<i>x</i>) = <i>x</i><sup>4</sup> − 10<i>x</i><sup>2</sup> + 1 = (<i>x</i> − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>)(<i>x</i> + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>)(<i>x</i> − <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>)(<i>x</i> + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span> + <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span>). </p><p>Notice if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ={\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6b60714c5ecd52dee7f5057b8b72b01b07e1969" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.684ex; height:3.009ex;" alt="{\displaystyle \alpha ={\sqrt {2}}}"></span> then the Galois action on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}"></span> stabilizes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. Hence the minimal polynomial can be found using the quotient group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gal</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gal</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30ec9771852c24c86bd5439e03d8f67760cc35b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.553ex; height:3.176ex;" alt="{\displaystyle {\text{Gal}}(\mathbb {Q} ({\sqrt {2}},{\sqrt {3}})/\mathbb {Q} )/{\text{Gal}}(\mathbb {Q} ({\sqrt {3}})/\mathbb {Q} )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Roots_of_unity">Roots of unity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=12" title="Edit section: Roots of unity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The minimal polynomials in <b>Q</b>[<i>x</i>] of <a href="/wiki/Root_of_unity" title="Root of unity">roots of unity</a> are the <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomials</a>. The roots of the <a href="/wiki/Minimal_polynomial_of_2cos(2pi/n)" title="Minimal polynomial of 2cos(2pi/n)">minimal polynomial of 2cos(2<span class="texhtml mvar" style="font-style:italic;">π</span>/n)</a> are twice the real part of the primitive roots of unity. </p> <div class="mw-heading mw-heading3"><h3 id="Swinnerton-Dyer_polynomials">Swinnerton-Dyer polynomials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=13" title="Edit section: Swinnerton-Dyer polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The minimal polynomial in <b>Q</b>[<i>x</i>] of the sum of the square roots of the first <i>n</i> prime numbers is constructed analogously, and is called a <a href="/wiki/Swinnerton-Dyer_polynomial" title="Swinnerton-Dyer polynomial">Swinnerton-Dyer polynomial</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></li> <li><a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Minimal_polynomial_(field_theory)&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Algebraic_Number_Minimal_Polynomial"><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 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(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/AlgebraicNumberMinimalPolynomial.html">"Algebraic Number Minimal Polynomial"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Algebraic+Number+Minimal+Polynomial&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FAlgebraicNumberMinimalPolynomial.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMinimal+polynomial+%28field+theory%29" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/MinimalPolynomial">Minimal polynomial</a> at <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>.</li> <li>Pinter, Charles C. <i>A Book of Abstract Algebra</i>. Dover Books on Mathematics Series. Dover Publications, 2010, p. 270–273. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47417-5" title="Special:BookSources/978-0-486-47417-5">978-0-486-47417-5</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Minimal_polynomial_(field_theory)&oldid=1257532015">https://en.wikipedia.org/w/index.php?title=Minimal_polynomial_(field_theory)&oldid=1257532015</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:Polynomials" title="Category:Polynomials">Polynomials</a></li><li><a href="/wiki/Category:Field_(mathematics)" title="Category:Field (mathematics)">Field (mathematics)</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:Use_American_English_from_March_2019" title="Category:Use American English from March 2019">Use American English from March 2019</a></li><li><a href="/wiki/Category:All_Wikipedia_articles_written_in_American_English" title="Category:All Wikipedia articles written in American English">All Wikipedia articles written in American English</a></li><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_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</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 15 November 2024, at 11:57<span class="anonymous-show"> (UTC)</span>.</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>; additional terms may apply. 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