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Eisenstein integer - Wikipedia
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class="vector-toc-numb">2</span> <span>Euclidean domain</span> </div> </a> <ul id="toc-Euclidean_domain-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eisenstein_primes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eisenstein_primes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Eisenstein primes</span> </div> </a> <ul id="toc-Eisenstein_primes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eisenstein_series" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eisenstein_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Eisenstein series</span> </div> </a> <ul id="toc-Eisenstein_series-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quotient_of_C_by_the_Eisenstein_integers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Quotient_of_C_by_the_Eisenstein_integers"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Quotient of <span><b>C</b></span> by the Eisenstein integers</span> </div> </a> <ul id="toc-Quotient_of_C_by_the_Eisenstein_integers-sublist" class="vector-toc-list"> </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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" 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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 – Catalan" lang="ca" hreflang="ca" data-title="Enter d'Eisenstein" 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/Eisensteinovo_%C4%8D%C3%ADslo" title="Eisensteinovo číslo – Czech" lang="cs" hreflang="cs" data-title="Eisensteinovo číslo" 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/Eisenstein-Zahl" title="Eisenstein-Zahl – German" lang="de" hreflang="de" data-title="Eisenstein-Zahl" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Entero_de_Eisenstein" title="Entero de Eisenstein – Spanish" lang="es" hreflang="es" data-title="Entero de Eisenstein" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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 – French" lang="fr" hreflang="fr" data-title="Entier d'Eisenstein" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</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 – Korean" lang="ko" hreflang="ko" data-title="아이젠슈타인 정수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Italian" lang="it" hreflang="it" data-title="Intero di Eisenstein" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="חוג השלמים של אייזנשטיין" data-language-autonym="עברית" data-language-local-name="Hebrew" 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 – Hungarian" lang="hu" hreflang="hu" data-title="Eisenstein-egész" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Ајзенштајнов цел број" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Dutch" lang="nl" hreflang="nl" data-title="Geheel getal van Eisenstein" 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href="https://pt.wikipedia.org/wiki/Inteiro_de_Eisenstein" title="Inteiro de Eisenstein – Portuguese" lang="pt" hreflang="pt" data-title="Inteiro de Eisenstein" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A7%D0%B8%D1%81%D0%BB%D0%BE_%D0%AD%D0%B9%D0%B7%D0%B5%D0%BD%D1%88%D1%82%D0%B5%D0%B9%D0%BD%D0%B0" title="Число Эйзенштейна – Russian" lang="ru" hreflang="ru" data-title="Число Эйзенштейна" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Eisensteinin_kokonaisluku" title="Eisensteinin kokonaisluku – Finnish" lang="fi" hreflang="fi" data-title="Eisensteinin kokonaisluku" data-language-autonym="Suomi" 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class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Eisenstein_prime&redirect=no" class="mw-redirect" title="Eisenstein prime">Eisenstein prime</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">"Eulerian integer" and "Euler integer" redirect here. For other uses, see <a href="/wiki/List_of_topics_named_after_Leonhard_Euler#Euler's_numbers" title="List of topics named after Leonhard Euler">List of topics named after Leonhard Euler § Euler's numbers</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Complex number whose mapping on a coordinate plane produces a triangular lattice</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Eisenstein_integer" title="Special:EditPage/Eisenstein integer">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Eisenstein+integer%22">"Eisenstein integer"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Eisenstein+integer%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Eisenstein+integer%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Eisenstein+integer%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Eisenstein+integer%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Eisenstein+integer%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">July 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>Eisenstein integers</b> (named after <a href="/wiki/Gotthold_Eisenstein" title="Gotthold Eisenstein">Gotthold Eisenstein</a>), occasionally also known<sup id="cite_ref-euler-name_1-0" class="reference"><a href="#cite_note-euler-name-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> as <b>Eulerian integers</b> (after <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>), are the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> of the form </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> <mi>ω<!-- ω --></mi> <mo>,</mo> </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/7c0d7a195d722a319034277da7bef081a5f2527e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.347ex; height:2.509ex;" alt="{\displaystyle z=a+b\omega ,}"></span></dd></dl> <p>where <span class="texhtml"><i>a</i></span> and <span class="texhtml"><i>b</i></span> are <a href="/wiki/Integer" title="Integer">integers</a> and </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+i{\sqrt {3}}}{2}}=e^{i2\pi /3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω<!-- ω --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−<!-- − --></mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π<!-- π --></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+i{\sqrt {3}}}{2}}=e^{i2\pi /3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ee19db168eb48ff977b7a7ad49dedeb6214882" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.482ex; height:5.843ex;" alt="{\displaystyle \omega ={\frac {-1+i{\sqrt {3}}}{2}}=e^{i2\pi /3}}"></span></dd></dl> <p>is a <a href="/wiki/Root_of_unity#General_definition" title="Root of unity">primitive</a> (hence non-real) <a href="/wiki/Cube_root_of_unity" class="mw-redirect" title="Cube root of unity">cube root of unity</a>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Eisenstein_integer_lattice.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Eisenstein_integer_lattice.png/191px-Eisenstein_integer_lattice.png" decoding="async" width="191" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Eisenstein_integer_lattice.png/287px-Eisenstein_integer_lattice.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a8/Eisenstein_integer_lattice.png/382px-Eisenstein_integer_lattice.png 2x" data-file-width="383" data-file-height="241" /></a><figcaption>Eisenstein integers as the points of a certain triangular lattice in the complex plane</figcaption></figure> <p>The Eisenstein integers form a <a href="/wiki/Triangular_lattice" class="mw-redirect" title="Triangular lattice">triangular lattice</a> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>, in contrast with the <a href="/wiki/Gaussian_integers" class="mw-redirect" title="Gaussian integers">Gaussian integers</a>, which form a <a href="/wiki/Square_lattice" title="Square lattice">square lattice</a> in the complex plane. The Eisenstein integers are a <a href="/wiki/Countable_set" title="Countable set">countably infinite set</a>. </p> <meta property="mw:PageProp/toc" /> <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=Eisenstein_integer&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Eisenstein integers form a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> of <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a> in the <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number field</a> <span class="texhtml"><b>Q</b>(<i>ω</i>)</span> – the third <a href="/wiki/Cyclotomic_field" title="Cyclotomic field">cyclotomic field</a>. To see that the Eisenstein integers are algebraic integers note that each <span class="texhtml"><i>z</i> = <i>a</i> + <i>bω</i></span> is a root of the <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{2}-(2a-b)\;\!z+\left(a^{2}-ab+b^{2}\right)~.}"> <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> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>z</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <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> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{2}-(2a-b)\;\!z+\left(a^{2}-ab+b^{2}\right)~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e92694a993f7716419f6499ff5b182257a3da967" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.199ex; height:3.343ex;" alt="{\displaystyle z^{2}-(2a-b)\;\!z+\left(a^{2}-ab+b^{2}\right)~.}"></span></dd></dl> <p>In particular, <span class="texhtml"><i>ω</i></span> satisfies the equation </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 ^{2}+\omega +1=0~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω<!-- ω --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>ω<!-- ω --></mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}+\omega +1=0~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb1e2b65cc415340d0457057d92de77338149e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.278ex; height:2.843ex;" alt="{\displaystyle \omega ^{2}+\omega +1=0~.}"></span></dd></dl> <p>The product of two Eisenstein integers <span class="texhtml"><i>a</i> + <i>bω</i></span> and <span class="texhtml mvar" style="font-style:italic;"><i>c</i> + <i>dω</i></span> is given explicitly by </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+b\;\!\omega )\;\!(c+d\;\!\omega )=(ac-bd)+(bc+ad-bd)\;\!\omega ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>ω<!-- ω --></mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>c</mi> <mo>+</mo> <mi>d</mi> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>ω<!-- ω --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>c</mi> <mo>−<!-- − --></mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo>+</mo> <mi>a</mi> <mi>d</mi> <mo>−<!-- − --></mo> <mi>b</mi> <mi>d</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>ω<!-- ω --></mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b\;\!\omega )\;\!(c+d\;\!\omega )=(ac-bd)+(bc+ad-bd)\;\!\omega ~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d895327ef082d79e7a8768f9bbae6519552ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.539ex; height:2.843ex;" alt="{\displaystyle (a+b\;\!\omega )\;\!(c+d\;\!\omega )=(ac-bd)+(bc+ad-bd)\;\!\omega ~.}"></span></dd></dl> <p>The 2-norm of an Eisenstein integer is just its <a href="/wiki/Squared_modulus" class="mw-redirect" title="Squared modulus">squared modulus</a>, and is given by </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 {\left|a+b\;\!\omega \right|}^{2}\,=\,{(a-{\tfrac {1}{2}}b)}^{2}+{\tfrac {3}{4}}b^{2}\,=\,a^{2}-ab+b^{2}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mspace width="thickmathspace" /> <mspace width="negativethinmathspace" /> <mi>ω<!-- ω --></mi> </mrow> <mo>|</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <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> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\left|a+b\;\!\omega \right|}^{2}\,=\,{(a-{\tfrac {1}{2}}b)}^{2}+{\tfrac {3}{4}}b^{2}\,=\,a^{2}-ab+b^{2}~,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad33a410c8ef870faa3328fc933125ff32de23c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:46.476ex; height:4.009ex;" alt="{\displaystyle {\left|a+b\;\!\omega \right|}^{2}\,=\,{(a-{\tfrac {1}{2}}b)}^{2}+{\tfrac {3}{4}}b^{2}\,=\,a^{2}-ab+b^{2}~,}"></span></dd></dl> <p>which is clearly a positive ordinary (rational) integer. </p><p>Also, the <a href="/wiki/Complex_conjugation" class="mw-redirect" title="Complex conjugation">complex conjugate</a> of <span class="texhtml"><i>ω</i></span> satisfies </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 {\bar {\omega }}=\omega ^{2}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ω<!-- ω --></mi> <mo stretchy="false">¯<!-- ¯ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <msup> <mi>ω<!-- ω --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {\omega }}=\omega ^{2}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef8dcccdbc9f08c4f83415ca41c821dcd96ad74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.272ex; height:2.676ex;" alt="{\displaystyle {\bar {\omega }}=\omega ^{2}~.}"></span></dd></dl> <p>The <a href="/wiki/Group_of_units" class="mw-redirect" title="Group of units">group of units</a> in this ring is the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> formed by the sixth <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> in the complex plane: <span class="texhtml">{±1, ±<i>ω</i>, ±<i>ω</i><sup>2</sup>}</span>, the Eisenstein integers of norm <span class="texhtml">1</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Euclidean_domain">Euclidean domain</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eisenstein_integer&action=edit&section=2" title="Edit section: Euclidean domain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ring of Eisenstein integers forms a <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a> whose norm <span class="texhtml"><i>N</i></span> is given by the square modulus, as above: </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+b\,\omega )=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>b</mi> <mspace width="thinmathspace" /> <mi>ω<!-- ω --></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+b\,\omega )=a^{2}-ab+b^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1c30fbbd01739fc97ada7c96723088f01c56dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.763ex; height:3.176ex;" alt="{\displaystyle N(a+b\,\omega )=a^{2}-ab+b^{2}.}"></span></dd></dl> <p>A <a href="/wiki/Division_algorithm" title="Division algorithm">division algorithm</a>, applied to any dividend <span class="texhtml"><i>α</i></span> and divisor <span class="texhtml"><i>β</i> ≠ 0</span>, gives a quotient <span class="texhtml"><i>κ</i></span> and a remainder <span class="texhtml"><i>ρ</i></span> smaller than the divisor, satisfying: </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 \alpha =\kappa \beta +\rho \ \ {\text{ with }}\ \ N(\rho )<N(\beta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mi>κ<!-- κ --></mi> <mi>β<!-- β --></mi> <mo>+</mo> <mi>ρ<!-- ρ --></mi> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> with </mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ρ<!-- ρ --></mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>N</mi> <mo stretchy="false">(</mo> <mi>β<!-- β --></mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\kappa \beta +\rho \ \ {\text{ with }}\ \ N(\rho )<N(\beta ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79902138495fd4f6d38a1210793a7eb8fa5c0da1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.33ex; height:2.843ex;" alt="{\displaystyle \alpha =\kappa \beta +\rho \ \ {\text{ with }}\ \ N(\rho )<N(\beta ).}"></span></dd></dl> <p>Here, <span class="texhtml"><i>α</i></span>, <span class="texhtml"><i>β</i></span>, <span class="texhtml"><i>κ</i></span>, <span class="texhtml"><i>ρ</i></span> are all Eisenstein integers. This algorithm implies the <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a>, which proves <a href="/wiki/Euclid%27s_lemma" title="Euclid's lemma">Euclid's lemma</a> and the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">unique factorization</a> of Eisenstein integers into Eisenstein primes. </p><p>One division algorithm is as follows. First perform the division in the field of complex numbers, and write the quotient in terms of <span class="texhtml"><i>ω</i></span>: </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 {\frac {\alpha }{\beta }}\ =\ {\tfrac {1}{\ |\beta |^{2}}}\alpha {\overline {\beta }}\ =\ a+bi\ =\ a+{\tfrac {1}{\sqrt {3}}}b+{\tfrac {2}{\sqrt {3}}}b\omega ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α<!-- α --></mi> <mi>β<!-- β --></mi> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β<!-- β --></mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mstyle> </mrow> <mi>α<!-- α --></mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>β<!-- β --></mi> <mo accent="false">¯<!-- ¯ --></mo> </mover> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mstyle> </mrow> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mstyle> </mrow> <mi>b</mi> <mi>ω<!-- ω --></mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\alpha }{\beta }}\ =\ {\tfrac {1}{\ |\beta |^{2}}}\alpha {\overline {\beta }}\ =\ a+bi\ =\ a+{\tfrac {1}{\sqrt {3}}}b+{\tfrac {2}{\sqrt {3}}}b\omega ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84b0119ed956995bc6f03d603fb163fddda4dfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.076ex; height:5.176ex;" alt="{\displaystyle {\frac {\alpha }{\beta }}\ =\ {\tfrac {1}{\ |\beta |^{2}}}\alpha {\overline {\beta }}\ =\ a+bi\ =\ a+{\tfrac {1}{\sqrt {3}}}b+{\tfrac {2}{\sqrt {3}}}b\omega ,}"></span></dd></dl> <p>for rational <span class="texhtml"><i>a</i>, <i>b</i> ∈ <b>Q</b></span>. Then obtain the Eisenstein integer quotient by rounding the rational coefficients to the nearest integer: </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 \kappa =\left\lfloor a+{\tfrac {1}{\sqrt {3}}}b\right\rceil +\left\lfloor {\tfrac {2}{\sqrt {3}}}b\right\rceil \omega \ \ {\text{ and }}\ \ \rho ={\alpha }-\kappa \beta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ<!-- κ --></mi> <mo>=</mo> <mrow> <mo>⌊</mo> <mrow> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mstyle> </mrow> <mi>b</mi> </mrow> <mo>⌉</mo> </mrow> <mo>+</mo> <mrow> <mo>⌊</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mstyle> </mrow> <mi>b</mi> </mrow> <mo>⌉</mo> </mrow> <mi>ω<!-- ω --></mi> <mtext> </mtext> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> </mrow> <mo>−<!-- − --></mo> <mi>κ<!-- κ --></mi> <mi>β<!-- β --></mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa =\left\lfloor a+{\tfrac {1}{\sqrt {3}}}b\right\rceil +\left\lfloor {\tfrac {2}{\sqrt {3}}}b\right\rceil \omega \ \ {\text{ and }}\ \ \rho ={\alpha }-\kappa \beta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b573351d2727d54fbadc4de83d01c3510b4defa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:45.318ex; height:4.843ex;" alt="{\displaystyle \kappa =\left\lfloor a+{\tfrac {1}{\sqrt {3}}}b\right\rceil +\left\lfloor {\tfrac {2}{\sqrt {3}}}b\right\rceil \omega \ \ {\text{ and }}\ \ \rho ={\alpha }-\kappa \beta .}"></span></dd></dl> <p>Here <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 \lfloor x\rceil }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊<!-- ⌊ --></mo> <mi>x</mi> <mo fence="false" stretchy="false">⌉<!-- ⌉ --></mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor x\rceil }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d85f3c359bd3e3aee2a21e06fe49b7bf847dcaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.394ex; height:2.843ex;" alt="{\displaystyle \lfloor x\rceil }"></span> may denote any of the standard <a href="/wiki/Rounding" title="Rounding">rounding</a>-to-integer functions. </p><p>The reason this satisfies <span class="texhtml"><i>N</i>(<i>ρ</i>) < <i>N</i>(<i>β</i>)</span>, while the analogous procedure fails for most other <a href="/wiki/Quadratic_integer" title="Quadratic integer">quadratic integer</a> rings, is as follows. A fundamental domain for the ideal <span class="texhtml"><b>Z</b>[<i>ω</i>]<i>β</i> = <b>Z</b><i>β</i> + <b>Z</b><i>ωβ</i></span>, acting by translations on the complex plane, is the 60°–120° rhombus with vertices <span class="texhtml">0</span>, <span class="texhtml"><i>β</i></span>, <span class="texhtml"><i>ωβ</i></span>, <span class="texhtml"><i>β</i> + <i>ωβ</i></span>. Any Eisenstein integer <span class="texhtml"><i>α</i></span> lies inside one of the translates of this parallelogram, and the quotient <span class="texhtml"><i>κ</i></span> is one of its vertices. The remainder is the square distance from <span class="texhtml"><i>α</i></span> to this vertex, but the maximum possible distance in our algorithm is only <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 {\tfrac {\sqrt {3}}{2}}|\beta |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {\sqrt {3}}{2}}|\beta |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efad8bd90707dd90a637717491dd162ae9c9ad39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.653ex; height:4.176ex;" alt="{\displaystyle {\tfrac {\sqrt {3}}{2}}|\beta |}"></span>, so <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 |\rho |\leq {\tfrac {\sqrt {3}}{2}}|\beta |<|\beta |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤<!-- ≤ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo><</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>β<!-- β --></mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\rho |\leq {\tfrac {\sqrt {3}}{2}}|\beta |<|\beta |}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3a57d413f8884ae2db5a1df20322117311ec03c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.971ex; height:4.176ex;" alt="{\displaystyle |\rho |\leq {\tfrac {\sqrt {3}}{2}}|\beta |<|\beta |}"></span>. (The size of <span class="texhtml"><i>ρ</i></span> could be slightly decreased by taking <span class="texhtml"><i>κ</i></span> to be the closest corner.) </p> <div class="mw-heading mw-heading2"><h2 id="Eisenstein_primes">Eisenstein primes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eisenstein_integer&action=edit&section=3" title="Edit section: Eisenstein primes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For the unrelated concept of an Eisenstein prime of a modular curve, see <a href="/wiki/Eisenstein_ideal" title="Eisenstein ideal">Eisenstein ideal</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:EisensteinPrimes-01.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/EisensteinPrimes-01.svg/330px-EisensteinPrimes-01.svg.png" decoding="async" width="330" height="287" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/EisensteinPrimes-01.svg/495px-EisensteinPrimes-01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/EisensteinPrimes-01.svg/660px-EisensteinPrimes-01.svg.png 2x" data-file-width="360" data-file-height="313" /></a><figcaption>Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form <span class="texhtml">3<i>n</i> + 2</span>. All others have an absolute value equal to 3 or square root of a natural prime of the form <span class="texhtml">3<i>n</i> + 1</span>.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Eisenstein_primes.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Eisenstein_primes.svg/330px-Eisenstein_primes.svg.png" decoding="async" width="330" height="326" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Eisenstein_primes.svg/495px-Eisenstein_primes.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Eisenstein_primes.svg/660px-Eisenstein_primes.svg.png 2x" data-file-width="1203" data-file-height="1188" /></a><figcaption>Eisenstein primes in a larger range</figcaption></figure> <p>If <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>y</i></span> are Eisenstein integers, we say that <span class="texhtml"><i>x</i></span> divides <span class="texhtml"><i>y</i></span> if there is some Eisenstein integer <span class="texhtml"><i>z</i></span> such that <span class="texhtml"><i>y</i> = <i>zx</i></span>. A non-unit Eisenstein integer <span class="texhtml"><i>x</i></span> is said to be an Eisenstein prime if its only non-unit divisors are of the form <span class="texhtml"><i>ux</i></span>, where <span class="texhtml"><i>u</i></span> is any of the six units. They are the corresponding concept to the <a href="/wiki/Gaussian_prime" class="mw-redirect" title="Gaussian prime">Gaussian primes</a> in the Gaussian integers. </p><p>There are two types of Eisenstein prime. </p> <ul><li>an ordinary <a href="/wiki/Prime_number" title="Prime number">prime number</a> (or <i>rational prime</i>) which is congruent to <span class="texhtml">2 mod 3</span> is also an Eisenstein prime.</li> <li><span class="texhtml">3</span> and each rational prime congruent to <span class="texhtml">1 mod 3</span> are equal to the norm <span class="texhtml"><i>x</i><sup>2</sup> − <i>xy</i> + <i>y</i><sup>2</sup></span> of an Eisenstein integer <span class="texhtml"><i>x</i> + <i>ωy</i></span>. Thus, such a prime may be factored as <span class="texhtml">(<i>x</i> + <i>ωy</i>)(<i>x</i> + <i>ω</i><sup>2</sup><i>y</i>)</span>, and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.</li></ul> <p>In the second type, factors of <span class="texhtml">3</span>, <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 1-\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ω<!-- ω --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3608960c6864aaa2da1347dd0b2fb90372697b36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.449ex; height:2.343ex;" alt="{\displaystyle 1-\omega }"></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 1-\omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>ω<!-- ω --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\omega ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33fac0e6210f7aa206608773156ba81f88fe32f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.503ex; height:2.843ex;" alt="{\displaystyle 1-\omega ^{2}}"></span> are <a href="/wiki/Associate_(ring_theory)" class="mw-redirect" title="Associate (ring theory)">associates</a>: <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 1-\omega =(-\omega )(1-\omega ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−<!-- − --></mo> <mi>ω<!-- ω --></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−<!-- − --></mo> <mi>ω<!-- ω --></mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−<!-- − --></mo> <msup> <mi>ω<!-- ω --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-\omega =(-\omega )(1-\omega ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6465ecb54e1f1d4ec9cc10b484f1ab04e460b569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.923ex; height:3.176ex;" alt="{\displaystyle 1-\omega =(-\omega )(1-\omega ^{2})}"></span>, so it is regarded as a special type in some books.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The first few Eisenstein primes of the form <span class="texhtml">3<i>n</i> − 1</span> are: </p> <dl><dd><a href="/wiki/2_(number)" class="mw-redirect" title="2 (number)">2</a>, <a href="/wiki/5_(number)" class="mw-redirect" title="5 (number)">5</a>, <a href="/wiki/11_(number)" title="11 (number)">11</a>, <a href="/wiki/17_(number)" title="17 (number)">17</a>, <a href="/wiki/23_(number)" title="23 (number)">23</a>, <a href="/wiki/29_(number)" title="29 (number)">29</a>, <a href="/wiki/41_(number)" title="41 (number)">41</a>, <a href="/wiki/47_(number)" title="47 (number)">47</a>, <a href="/wiki/53_(number)" title="53 (number)">53</a>, <a href="/wiki/59_(number)" title="59 (number)">59</a>, <a href="/wiki/71_(number)" title="71 (number)">71</a>, <a href="/wiki/83_(number)" title="83 (number)">83</a>, <a href="/wiki/89_(number)" title="89 (number)">89</a>, <a href="/wiki/101_(number)" title="101 (number)">101</a>, ... (sequence <span class="nowrap external"><a href="//oeis.org/A003627" class="extiw" title="oeis:A003627">A003627</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>).</dd></dl> <p>Natural primes that are congruent to <span class="texhtml">0</span> or <span class="texhtml">1</span> modulo <span class="texhtml">3</span> are <i>not</i> Eisenstein primes:<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> they admit nontrivial factorizations in <span class="texhtml"><b>Z</b>[<i>ω</i>]</span>. For example: </p> <dl><dd><span class="texhtml">3 = −(1 + 2<i>ω</i>)<sup>2</sup></span></dd> <dd><span class="texhtml">7 = (3 + <i>ω</i>)(2 − <i>ω</i>)</span>.</dd></dl> <p>In general, if a natural prime <span class="texhtml"><i>p</i></span> is <span class="texhtml">1</span> modulo <span class="texhtml">3</span> and can therefore be written as <span class="texhtml"><i>p</i> = <i>a</i><sup>2</sup> − <i>ab</i> + <i>b</i><sup>2</sup></span>, then it factorizes over <span class="texhtml"><b>Z</b>[<i>ω</i>]</span> as </p> <dl><dd><span class="texhtml">p = (<i>a</i> + <i>bω</i>)((<i>a</i> − <i>b</i>) − <i>bω</i>)</span>.</dd></dl> <p>Some non-real Eisenstein primes are </p> <dl><dd><span class="texhtml">2 + <i>ω</i></span>, <span class="texhtml">3 + <i>ω</i></span>, <span class="texhtml">4 + <i>ω</i></span>, <span class="texhtml">5 + 2<i>ω</i></span>, <span class="texhtml">6 + <i>ω</i></span>, <span class="texhtml">7 + <i>ω</i></span>, <span class="texhtml">7 + 3<i>ω</i></span>.</dd></dl> <p>Up to conjugacy and unit multiples, the primes listed above, together with <span class="texhtml">2</span> and <span class="texhtml">5</span>, are all the Eisenstein primes of <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> not exceeding <span class="texhtml">7</span>. </p><p>As of October 2023<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Eisenstein_integer&action=edit">[update]</a></sup>, the largest known real Eisenstein prime is the <a href="/wiki/Largest_known_prime_number" title="Largest known prime number">tenth-largest known prime</a> <span class="texhtml">10223 × 2<sup>31172165</sup> + 1</span>, discovered by Péter Szabolcs and <a href="/wiki/PrimeGrid" title="PrimeGrid">PrimeGrid</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> With one exception,<sup class="noprint Inline-Template" style="margin-left:0.1em; white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The recently added 7th-largest prime ≡ 2 (mod 3); this likely makes it the largest known real Eisenstein prime. (October 2023)">clarification needed</span></a></i>]</sup> all larger known primes are <a href="/wiki/Mersenne_prime" title="Mersenne prime">Mersenne primes</a>, discovered by <a href="/wiki/GIMPS" class="mw-redirect" title="GIMPS">GIMPS</a>. Real Eisenstein primes are congruent to <span class="texhtml">2 mod 3</span>, and all Mersenne primes greater than <span class="texhtml">3</span> are congruent to <span class="texhtml">1 mod 3</span>; thus no Mersenne prime is an Eisenstein prime. </p> <div class="mw-heading mw-heading2"><h2 id="Eisenstein_series">Eisenstein series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eisenstein_integer&action=edit&section=4" title="Edit section: Eisenstein series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sum of the reciprocals of all Eisenstein integers excluding <span class="texhtml">0</span> raised to the fourth power is <span class="texhtml">0</span>:<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}\left(e^{\frac {2\pi i}{3}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π<!-- π --></mi> <mi>i</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}\left(e^{\frac {2\pi i}{3}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40c933b656acc41512d05529d3ec250ef464fdc8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:27.857ex; height:7.176ex;" alt="{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{4}}}=G_{4}\left(e^{\frac {2\pi i}{3}}\right)=0}"></span> so <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 e^{2\pi i/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π<!-- 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 e^{2\pi i/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d898e757a3746aebce946a993dbb332497f0521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.291ex; height:2.843ex;" alt="{\displaystyle e^{2\pi i/3}}"></span> is a root of <a href="/wiki/J-invariant" title="J-invariant">j-invariant</a>. In general <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_{k}\left(e^{\frac {2\pi i}{3}}\right)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π<!-- π --></mi> <mi>i</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{k}\left(e^{\frac {2\pi i}{3}}\right)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08734d98bc955d85b4285f4755718c14b27cf415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.029ex; height:6.176ex;" alt="{\displaystyle G_{k}\left(e^{\frac {2\pi i}{3}}\right)=0}"></span> if and only 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 k\not \equiv 0{\pmod {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>≢</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\not \equiv 0{\pmod {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8809de706d5f950da816ff4f1ee9e2368cba9ca8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.32ex; height:2.843ex;" alt="{\displaystyle k\not \equiv 0{\pmod {6}}}"></span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The sum of the reciprocals of all Eisenstein integers excluding <span class="texhtml">0</span> raised to the sixth power can be expressed in terms of the <a href="/wiki/Gamma_function" title="Gamma function">gamma function</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{6}}}=G_{6}\left(e^{\frac {2\pi i}{3}}\right)={\frac {\Gamma (1/3)^{18}}{8960\pi ^{6}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo class="MJX-variant">∖<!-- ∖ --></mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π<!-- π --></mi> <mi>i</mi> </mrow> <mn>3</mn> </mfrac> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Γ<!-- Γ --></mi> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> </mrow> <mrow> <mn>8960</mn> <msup> <mi>π<!-- π --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{6}}}=G_{6}\left(e^{\frac {2\pi i}{3}}\right)={\frac {\Gamma (1/3)^{18}}{8960\pi ^{6}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c024bc79a02ed6a327214b82c8ec17599bec1999" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:36.156ex; height:7.509ex;" alt="{\displaystyle \sum _{z\in \mathbf {E} \setminus \{0\}}{\frac {1}{z^{6}}}=G_{6}\left(e^{\frac {2\pi i}{3}}\right)={\frac {\Gamma (1/3)^{18}}{8960\pi ^{6}}}}"></span> where <span class="texhtml"><b>E</b></span> are the Eisenstein integers and <span class="texhtml"><i>G</i><sub>6</sub></span> is the <a href="/wiki/Eisenstein_series" title="Eisenstein series">Eisenstein series</a> of weight 6.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Quotient_of_C_by_the_Eisenstein_integers">Quotient of <span class="texhtml"><b>C</b></span> by the Eisenstein integers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eisenstein_integer&action=edit&section=5" title="Edit section: Quotient of C by the Eisenstein integers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Quotient_ring" title="Quotient ring">quotient</a> of the complex plane <span class="texhtml"><b>C</b></span> by the <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> containing all Eisenstein integers is a <a href="/wiki/Complex_torus" title="Complex torus">complex torus</a> of real dimension <span class="texhtml">2</span>. This is one of two tori with maximal <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> among all such complex tori.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2013)">citation needed</span></a></i>]</sup> This torus can be obtained by identifying each of the three pairs of opposite edges of a regular hexagon. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Wrapped_hexagon_topology.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Wrapped_hexagon_topology.png/220px-Wrapped_hexagon_topology.png" decoding="async" width="220" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Wrapped_hexagon_topology.png/330px-Wrapped_hexagon_topology.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Wrapped_hexagon_topology.png/440px-Wrapped_hexagon_topology.png 2x" data-file-width="656" data-file-height="540" /></a><figcaption>Identifying each of the three pairs of opposite edges of a regular hexagon.</figcaption></figure> <p>The other maximally symmetric torus is the quotient of the complex plane by the additive lattice of <a href="/wiki/Gaussian_integers" class="mw-redirect" title="Gaussian integers">Gaussian integers</a>, and can be obtained by identifying each of the two pairs of opposite sides of a square fundamental domain, such as <span class="texhtml">[0, 1] × [0, 1]</span>. </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=Eisenstein_integer&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integer</a></li> <li><a href="/wiki/Cyclotomic_field" title="Cyclotomic field">Cyclotomic field</a></li> <li><a href="/wiki/Systolic_geometry" title="Systolic geometry">Systolic geometry</a></li> <li><a href="/wiki/Hermite_constant" title="Hermite constant">Hermite constant</a></li> <li><a href="/wiki/Cubic_reciprocity" title="Cubic reciprocity">Cubic reciprocity</a></li> <li><a href="/wiki/Loewner%27s_torus_inequality" title="Loewner's torus inequality">Loewner's torus inequality</a></li> <li><a href="/wiki/Hurwitz_quaternion" title="Hurwitz quaternion">Hurwitz quaternion</a></li> <li><a href="/wiki/Quadratic_integer" title="Quadratic integer">Quadratic integer</a></li> <li><a href="/wiki/Dixon_elliptic_functions" title="Dixon elliptic functions">Dixon elliptic functions</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eisenstein_integer&action=edit&section=7" title="Edit section: Notes"><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 class="mw-references-wrap"><ol class="references"> <li id="cite_note-euler-name-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-euler-name_1-0">^</a></b></span> <span class="reference-text">Both <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSurányi1997" class="citation book cs1">Surányi, László (1997). <i>Algebra</i>. TYPOTEX. p. 73.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=73&rft.pub=TYPOTEX&rft.date=1997&rft.aulast=Sur%C3%A1nyi&rft.aufirst=L%C3%A1szl%C3%B3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span> and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSzalay1991" class="citation book cs1">Szalay, Mihály (1991). <i>Számelmélet</i>. Tankönyvkiadó. p. 75.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sz%C3%A1melm%C3%A9let&rft.pages=75&rft.pub=Tank%C3%B6nyvkiad%C3%B3&rft.date=1991&rft.aulast=Szalay&rft.aufirst=Mih%C3%A1ly&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span> call these numbers "Euler-egészek", that is, Eulerian integers. The latter claims Euler worked with them in a proof.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Eisenstein_integer"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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/EisensteinPrime.html">"Eisenstein integer"</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=Eisenstein+integer&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FEisensteinPrime.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCox1997" class="citation book cs1">Cox, David A. (1997-05-08). <a rel="nofollow" class="external text" href="https://www.math.utoronto.ca/~ila/Cox-Primes_of_the_form_x2+ny2.pdf"><i>Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication</i></a> <span class="cs1-format">(PDF)</span>. Wiley. p. 77. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-19079-9" title="Special:BookSources/0-471-19079-9"><bdi>0-471-19079-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Primes+of+the+Form+x2%2Bny2%3A+Fermat%2C+Class+Field+Theory+and+Complex+Multiplication&rft.pages=77&rft.pub=Wiley&rft.date=1997-05-08&rft.isbn=0-471-19079-9&rft.aulast=Cox&rft.aufirst=David+A.&rft_id=https%3A%2F%2Fwww.math.utoronto.ca%2F~ila%2FCox-Primes_of_the_form_x2%2Bny2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/4242234/x2-x-1-is-reducible-in-mathbbf-p-x-iff-p-equiv-1-mod-3">"<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}+X+1}"> <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> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}+X+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a543a762d8c8d5d7e45200729003279332a68321" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.874ex; height:2.843ex;" alt="{\displaystyle X^{2}+X+1}"></span> is reducible 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 \mathbb {F} _{p}[X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}[X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eded0c154aa847807d7f992fdf6db7eb5a73476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.753ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} _{p}[X]}"></span> iff <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 p\equiv 1{\pmod {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>≡<!-- ≡ --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\equiv 1{\pmod {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7247837156162562a12d2e17454341da15c51075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.366ex; height:2.843ex;" alt="{\displaystyle p\equiv 1{\pmod {3}}}"></span>"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=MATH+RENDER+ERROR+is+reducible+in+MATH+RENDER+ERROR+iff+MATH+RENDER+ERROR&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F4242234%2Fx2-x-1-is-reducible-in-mathbbf-p-x-iff-p-equiv-1-mod-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://primes.utm.edu/top20/page.php?id=3">"Largest Known Primes"</a>. <i>The <a href="/wiki/Prime_Pages" class="mw-redirect" title="Prime Pages">Prime Pages</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Prime+Pages&rft.atitle=Largest+Known+Primes&rft_id=https%3A%2F%2Fprimes.utm.edu%2Ftop20%2Fpage.php%3Fid%3D3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/1231810/what-are-the-zeros-of-the-j-function">"What are the zeros of the j-function?"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+are+the+zeros+of+the+j-function%3F&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F1231810%2Fwhat-are-the-zeros-of-the-j-function&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/2044686/show-that-g-4i-neq-0-and-g-6-rho-neq-0-rho-e2-pi-i-3">"Show that <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_{4}(i)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{4}(i)\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2312d665eee99cf93343a98078df4cd9ac4b0bfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.754ex; height:2.843ex;" alt="{\displaystyle G_{4}(i)\neq 0}"></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 G_{6}(\rho )\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ρ<!-- ρ --></mi> <mo stretchy="false">)</mo> <mo>≠<!-- ≠ --></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{6}(\rho )\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c71ed52293a173731fe3bbf399aa43ba4d19292f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.153ex; height:2.843ex;" alt="{\displaystyle G_{6}(\rho )\neq 0}"></span>, <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 \rho =e^{2\pi i/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ<!-- ρ --></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π<!-- 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 \rho =e^{2\pi i/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ddd31ac8db66d1b62e2b6e15aa6c8f598690cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.591ex; height:3.343ex;" alt="{\displaystyle \rho =e^{2\pi i/3}}"></span>"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Show+that+MATH+RENDER+ERROR%2C+and+MATH+RENDER+ERROR%2C+MATH+RENDER+ERROR&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F2044686%2Fshow-that-g-4i-neq-0-and-g-6-rho-neq-0-rho-e2-pi-i-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://fungrim.org/entry/0fda1b/">"Entry 0fda1b – Fungrim: The Mathematical Functions Grimoire"</a>. <i>fungrim.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-06-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=fungrim.org&rft.atitle=Entry+0fda1b+%E2%80%93+Fungrim%3A+The+Mathematical+Functions+Grimoire&rft_id=https%3A%2F%2Ffungrim.org%2Fentry%2F0fda1b%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEisenstein+integer" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Eisenstein_integer&action=edit&section=8" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/EisensteinInteger.html">Eisenstein Integer--from MathWorld</a></li></ul> <div class="navbox-styles"><style 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