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<span>Relación con los primos de forma <i>x</i>² − <i>xy</i> + <i>y</i>²</span> </div> </a> <ul id="toc-Relación_con_los_primos_de_forma_x²_−_xy_+_y²-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Norma_de_un_entero_de_Eisenstein" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Norma_de_un_entero_de_Eisenstein"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Norma de un entero de Eisenstein</span> </div> </a> <button aria-controls="toc-Norma_de_un_entero_de_Eisenstein-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>Alternar subsección Norma de un entero de Eisenstein</span> </button> <ul id="toc-Norma_de_un_entero_de_Eisenstein-sublist" class="vector-toc-list"> <li id="toc-Otro_procedimiento" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" 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Disponible en 19 idiomas" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A3%D9%8A%D8%B2%D9%86%D8%B4%D8%AA%D8%A7%D9%8A%D9%86_%D8%A7%D9%84%D8%B5%D8%AD%D9%8A%D8%AD" title="عدد أيزنشتاين الصحيح (árabe)" lang="ar" hreflang="ar" data-title="عدد أيزنشتاين الصحيح" data-language-autonym="العربية" data-language-local-name="árabe" 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/Enter_d%27Eisenstein" title="Enter d'Eisenstein (catalán)" lang="ca" hreflang="ca" data-title="Enter d'Eisenstein" data-language-autonym="Català" data-language-local-name="catalán" 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/Eisensteinovo_%C4%8D%C3%ADslo" title="Eisensteinovo číslo (checo)" lang="cs" hreflang="cs" data-title="Eisensteinovo číslo" data-language-autonym="Čeština" data-language-local-name="checo" 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/Eisenstein-Zahl" title="Eisenstein-Zahl (alemán)" lang="de" hreflang="de" data-title="Eisenstein-Zahl" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Eisenstein_integer" title="Eisenstein integer (inglés)" lang="en" hreflang="en" data-title="Eisenstein integer" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Eisensteinin_kokonaisluku" title="Eisensteinin kokonaisluku (finés)" lang="fi" hreflang="fi" data-title="Eisensteinin kokonaisluku" data-language-autonym="Suomi" data-language-local-name="finés" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Entier_d%27Eisenstein" title="Entier d'Eisenstein (francés)" lang="fr" hreflang="fr" data-title="Entier d'Eisenstein" data-language-autonym="Français" data-language-local-name="francés" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%92_%D7%94%D7%A9%D7%9C%D7%9E%D7%99%D7%9D_%D7%A9%D7%9C_%D7%90%D7%99%D7%99%D7%96%D7%A0%D7%A9%D7%98%D7%99%D7%99%D7%9F" title="חוג השלמים של אייזנשטיין (hebreo)" lang="he" hreflang="he" data-title="חוג השלמים של אייזנשטיין" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Eisenstein-eg%C3%A9sz" title="Eisenstein-egész (húngaro)" lang="hu" hreflang="hu" data-title="Eisenstein-egész" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Intero_di_Eisenstein" title="Intero di Eisenstein (italiano)" lang="it" hreflang="it" data-title="Intero di Eisenstein" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%82%A4%E3%82%BC%E3%83%B3%E3%82%B7%E3%83%A5%E3%82%BF%E3%82%A4%E3%83%B3%E6%95%B4%E6%95%B0" title="アイゼンシュタイン整数 (japonés)" lang="ja" hreflang="ja" data-title="アイゼンシュタイン整数" data-language-autonym="日本語" data-language-local-name="japonés" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%EC%9D%B4%EC%A0%A0%EC%8A%88%ED%83%80%EC%9D%B8_%EC%A0%95%EC%88%98" title="아이젠슈타인 정수 (coreano)" lang="ko" hreflang="ko" data-title="아이젠슈타인 정수" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%90%D1%98%D0%B7%D0%B5%D0%BD%D1%88%D1%82%D0%B0%D1%98%D0%BD%D0%BE%D0%B2_%D1%86%D0%B5%D0%BB_%D0%B1%D1%80%D0%BE%D1%98" title="Ајзенштајнов цел број (macedonio)" lang="mk" hreflang="mk" data-title="Ајзенштајнов цел број" data-language-autonym="Македонски" data-language-local-name="macedonio" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Geheel_getal_van_Eisenstein" title="Geheel getal van Eisenstein (neerlandés)" lang="nl" hreflang="nl" data-title="Geheel getal van Eisenstein" data-language-autonym="Nederlands" data-language-local-name="neerlandés" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Eisenstein_integer_lattice.png/375px-Eisenstein_integer_lattice.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a8/Eisenstein_integer_lattice.png 2x" data-file-width="383" data-file-height="241" /></a><figcaption>Enteros de Eisenstein como puntos de intersección de una retícula triangular en el plano complejo.</figcaption></figure> <p>En <a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">matemáticas</a>, en especial en <a href="/wiki/Teor%C3%ADa_de_n%C3%BAmeros" title="Teoría de números">la teoría de números</a>, un <b>entero de Eisenstein</b>, llamado así en honor de <a href="/wiki/Ferdinand_Eisenstein" title="Ferdinand Eisenstein">Ferdinand Eisenstein</a>, es un <a href="/wiki/N%C3%BAmero_complejo" title="Número complejo">número complejo</a> de la forma </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=a+b\,\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=a+b\,\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89108fae9351364cbe3dd2369dc530afde9e3b56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.087ex; height:2.343ex;" alt="{\displaystyle z=a+b\,\omega }"></span></dd></dl> <p>donde <i>a</i> y <i>b</i> son <a href="/wiki/N%C3%BAmeros_enteros" class="mw-redirect" title="Números enteros">números enteros</a> y </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0963b96230b2bcaaf3b9206c40143d38eacbd1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.453ex; height:5.176ex;" alt="{\displaystyle \omega ={\frac {1}{2}}(-1+i{\sqrt {3}})=e^{2\pi i/3}}"></span></dd></dl> <p>es una de las <a href="/wiki/Ra%C3%ADz_de_la_unidad" title="Raíz de la unidad">raíces cúbicas imaginarias de 1</a> . </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Propiedades">Propiedades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=1" title="Editar sección: Propiedades"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Los enteros de Eisenstein forman un <a href="/wiki/Anillo_conmutativo" title="Anillo conmutativo">anillo conmutativo</a> de <a href="/wiki/N%C3%BAmero_algebraico" title="Número algebraico">enteros algebraicos</a> en el <a href="/wiki/Cuerpo_de_los_n%C3%BAmeros_algebraicos" class="mw-redirect" title="Cuerpo de los números algebraicos">cuerpo de los números algebraicos</a> <b>Q</b>(√−3). También forman un <a href="/wiki/Dominio_euclidiano" class="mw-redirect" title="Dominio euclidiano">dominio euclidiano</a>. </p><p>Para ver que los enteros de Eisenstein son enteros algebraicos nótese que cada <i>z</i> = <i>a</i> + <i>b</i>ω es un cero del <a href="/wiki/Polinomio_cuadr%C3%A1tico" class="mw-redirect" title="Polinomio cuadrático">polinomio cuadrático</a> de coeficiente principal = 1 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{2}-(2a-b)z+(a^{2}-ab+b^{2}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>z</mi> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{2}-(2a-b)z+(a^{2}-ab+b^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc8afa9d990864b93abc203b60c32eb0d5eb953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.653ex; height:3.176ex;" alt="{\displaystyle z^{2}-(2a-b)z+(a^{2}-ab+b^{2}).}"></span></dd></dl> <p>En particular, ω satisface la ecuación algebraica de coeficiente principal = 1; sus demás coeficientes son enteros racionales. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{2}+z+1=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>z</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{2}+z+1=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d6f88304f5a087319bb2115b11203472c3de76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.984ex; height:2.843ex;" alt="{\displaystyle z^{2}+z+1=0.}"></span></dd></dl> <p>Si <i>x</i> e <i>y</i> son enteros de Eisenstein, diremos que <i>x</i> <i>divide</i> a <i>y</i> si existe algún entero de Eisenstein <i>z</i> tal que </p> <dl><dd><i>y</i> = <i>z</i> <i>x</i>.</dd></dl> <p>Esto extiende la noción de <a href="/wiki/Divisibilidad" title="Divisibilidad">divisibilidad</a> para los enteros ordinarios, o sea los elementos del conjunto ℤ. Por lo tanto, podremos también extender la noción de <a href="/wiki/N%C3%BAmero_primo" title="Número primo">primalidad</a>; un entero de Eisenstein <i>x</i> será un <a href="/wiki/Primo_de_Eisenstein" title="Primo de Eisenstein">primo de Eisenstein</a> si sus únicos divisores son </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm x,\pm \omega x,\pm \omega ^{2}x,\pm 1,\pm \omega ,\pm \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mi>x</mi> <mo>,</mo> <mo>±</mo> <mi>ω</mi> <mi>x</mi> <mo>,</mo> <mo>±</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>ω</mi> <mo>,</mo> <mo>±</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm x,\pm \omega x,\pm \omega ^{2}x,\pm 1,\pm \omega ,\pm \omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5219911c137ddd12fddbf17dbf4404717ae8bfe3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.062ex; height:3.009ex;" alt="{\displaystyle \pm x,\pm \omega x,\pm \omega ^{2}x,\pm 1,\pm \omega ,\pm \omega ^{2}}"></span></dd></dl> <p>—excepto porque no consideraremos ±1, ±ω o ±ω² en sí mismos como primos de Eisenstein — son unidades en el <a href="/wiki/Anillo_(matem%C3%A1ticas)" class="mw-redirect" title="Anillo (matemáticas)">anillo</a> de los enteros de Eisenstein, y cada uno tiene norma = 1. </p> <div class="mw-heading mw-heading2"><h2 id="Relación_con_los_primos_de_forma_x²_−_xy_+_y²"><span id="Relaci.C3.B3n_con_los_primos_de_forma_x.C2.B2_.E2.88.92_xy_.2B_y.C2.B2"></span>Relación con los primos de forma <i>x</i>² − <i>xy</i> + <i>y</i>²</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=2" title="Editar sección: Relación con los primos de forma x² − xy + y²"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Puede demostrarse que un primo de la forma <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^{2}-xy+y^{2}}"> <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> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-xy+y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb50cdc830dcb9e68ac00169cec2c2036382969c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.765ex; height:3.009ex;" alt="{\displaystyle x^{2}-xy+y^{2}}"></span> puede ser factorizado en <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+\omega y)(x+\omega ^{2}y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>ω</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+\omega y)(x+\omega ^{2}y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b55135ac3c57a2c1f2400a8a408aff12553c3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.216ex; height:3.176ex;" alt="{\displaystyle (x+\omega y)(x+\omega ^{2}y)}"></span> y por lo tanto no es primo en el anillo de los enteros de Eisenstein. Nótese también que un número de la forma <i>x</i>² − <i>xy</i> + <i>y</i>² es primo si y solo si x + ωy es un primo de Eisenstein. </p> <div class="mw-heading mw-heading2"><h2 id="Norma_de_un_entero_de_Eisenstein">Norma de un entero de Eisenstein</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=3" title="Editar sección: Norma de un entero de Eisenstein"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>El anillo de los enteros de Eisenstein forma un <a href="/wiki/Dominio_euclidiano" class="mw-redirect" title="Dominio euclidiano">dominio euclidiano</a> cuya <a href="/wiki/Norma_(matem%C3%A1ticas)" class="mw-redirect" title="Norma (matemáticas)">norma</a> <i>N</i> es </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(a+\omega b)=a^{2}-ab+b^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>ω</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(a+\omega b)=a^{2}-ab+b^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f1898a26a8c95757cecdd96dacee49b36d00e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.376ex; height:3.176ex;" alt="{\displaystyle N(a+\omega b)=a^{2}-ab+b^{2}.}"></span> (1)</dd></dl> <p>Esto puede deducirse considerando los enteros de Eisenstein como números complejos: puesto que </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(a+ib)=a^{2}+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(a+ib)=a^{2}+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13f76f11e459504ac2e3a679280ed9ad50b3885b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.018ex; height:3.176ex;" alt="{\displaystyle N(a+ib)=a^{2}+b^{2}}"></span></dd></dl> <p>y puesto que </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+\omega b=\left(a-{1 \over 2}b\right)+i{{\sqrt {3}} \over 2}b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>ω</mi> <mi>b</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+\omega b=\left(a-{1 \over 2}b\right)+i{{\sqrt {3}} \over 2}b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f224398ce1d2bce568279fbd62e51781ac8097" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.674ex; height:6.509ex;" alt="{\displaystyle a+\omega b=\left(a-{1 \over 2}b\right)+i{{\sqrt {3}} \over 2}b}"></span></dd></dl> <p>se deduce que </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(a+\omega b)=\left(a-{1 \over 2}b\right)^{2}+{3 \over 4}b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>ω</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(a+\omega b)=\left(a-{1 \over 2}b\right)^{2}+{3 \over 4}b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/795033d2ca11568c24ea8823fa52e08972dc21e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.917ex; height:6.509ex;" alt="{\displaystyle N(a+\omega b)=\left(a-{1 \over 2}b\right)^{2}+{3 \over 4}b^{2}}"></span></dd></dl> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =a^{2}-ab+{1 \over 4}b^{2}+{3 \over 4}b^{2}=a^{2}-ab+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =a^{2}-ab+{1 \over 4}b^{2}+{3 \over 4}b^{2}=a^{2}-ab+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/262dfd55c83b2483e0ccdfd56cef75a7dafb3bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.929ex; height:5.176ex;" alt="{\displaystyle =a^{2}-ab+{1 \over 4}b^{2}+{3 \over 4}b^{2}=a^{2}-ab+b^{2}}"></span>.</dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Otro_procedimiento">Otro procedimiento</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=4" title="Editar sección: Otro procedimiento"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La norma de un entero de Eisenstein se puede definir como: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N(c+\omega d)=(c+\omega d)\times (c+\omega ^{2}d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>ω</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>ω</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(c+\omega d)=(c+\omega d)\times (c+\omega ^{2}d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0cd3053e2053b507213db505e180e2cc298feb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.011ex; height:3.176ex;" alt="{\displaystyle N(c+\omega d)=(c+\omega d)\times (c+\omega ^{2}d)}"></span></dd> <dd>pues se tiene <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(c+\omega d)=c^{2}+cd\omega +cd\omega ^{2}+\omega ^{3}d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>ω</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mi>ω</mi> <mo>+</mo> <mi>c</mi> <mi>d</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(c+\omega d)=c^{2}+cd\omega +cd\omega ^{2}+\omega ^{3}d^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a259a15c1a3e8a7a2c23ab0af0ce86633f475b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.226ex; height:3.176ex;" alt="{\displaystyle N(c+\omega d)=c^{2}+cd\omega +cd\omega ^{2}+\omega ^{3}d^{2}}"></span></dd> <dd>agrupando, teniendo en cuenta el cubo de omega, <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(c+\omega d)=c^{2}+cd(\omega +\omega ^{2})+1\times d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>ω</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>×</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(c+\omega d)=c^{2}+cd(\omega +\omega ^{2})+1\times d^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d8d5ec3d4fb985dd35399bc7efc51d6918ecff6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.316ex; height:3.176ex;" alt="{\displaystyle N(c+\omega d)=c^{2}+cd(\omega +\omega ^{2})+1\times d^{2}}"></span></dd> <dd>como <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 \omega +\omega ^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega +\omega ^{2}=-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cccc31f4757f3aaf0b6fa33c197ba59120b27755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.855ex; height:2.843ex;" alt="{\displaystyle \omega +\omega ^{2}=-1}"></span> resulta <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(c+\omega d)=c^{2}-cd+d^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>ω</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>c</mi> <mi>d</mi> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N(c+\omega d)=c^{2}-cd+d^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aa969454961d2556289d995417903cf1335ce65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.717ex; height:3.176ex;" alt="{\displaystyle N(c+\omega d)=c^{2}-cd+d^{2}}"></span>, lo mismo que (1).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Dominio_euclidiano">Dominio euclidiano</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=5" title="Editar sección: Dominio euclidiano"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dados dos enteros de Essenstein c y d ≠ 0, existen dos enteros de Essenstein q y r tal que </p> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=dq+r,\ N(d)>N(r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mi>d</mi> <mi>q</mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> <mtext> </mtext> <mi>N</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=dq+r,\ N(d)>N(r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2262f11541698c082cc85b145e9e33e84bf19dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.003ex; height:2.843ex;" alt="{\displaystyle c=dq+r,\ N(d)>N(r)}"></span> aunque q y r no sean únicos. Se sigue cumpliendo el algoritmo de Euclides.</dd></dl></dd></dl></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=6" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Primo_de_Eisenstein" title="Primo de Eisenstein">Primo de Eisenstein</a></li> <li><a href="/wiki/Entero_gaussiano" title="Entero gaussiano">Entero gaussiano</a></li> <li>Entero algebraico</li></ul> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entero_de_Eisenstein&action=edit&section=7" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>La versión inicial de este artículo es una adaptación de <a href="/w/index.php?title=Eisenstein_integer&action=edit&redlink=1" class="new" title="Eisenstein integer (aún no redactado)">Eisenstein integer</a> de <a href="/wiki/Wikipedia_en_ingl%C3%A9s" title="Wikipedia en inglés">Wikipedia en inglés</a> bajo licencia <a href="/wiki/GFDL" class="mw-redirect" title="GFDL">GFDL</a> y <a href="/wiki/Creative_Commons" title="Creative Commons">Creative Commons</a>.</i></li> <li><span id="Reference-Mathworld-Eisenstein_Integer" class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/EisensteinInteger.html">«Eisenstein Integer»</a>. 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