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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Enuntziatu baliokideak</span> </div> </a> <ul id="toc-Enuntziatu_baliokideak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Frogak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Frogak"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Frogak</span> </div> </a> <button aria-controls="toc-Frogak-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>Erakutsi/ezkutatu Frogak azpiatal</span> </button> <ul id="toc-Frogak-sublist" class="vector-toc-list"> <li id="toc-Liouville-ren_teorema_erabilita" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Liouville-ren_teorema_erabilita"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Liouville-ren teorema erabilita</span> </div> </a> <ul id="toc-Liouville-ren_teorema_erabilita-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Galois-en_teoria_erabilita" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Galois-en_teoria_erabilita"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Galois-en teoria erabilita</span> </div> </a> <ul id="toc-Galois-en_teoria_erabilita-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Korolarioak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Korolarioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Korolarioak</span> </div> </a> <ul id="toc-Korolarioak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Errefentziak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Errefentziak"> <div 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class="mw-content-ltr mw-parser-output" lang="eu" dir="ltr"><p><b>Aljebraren oinarrizko teoremaren</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> arabera, <b>d'Alembert-en teorema</b> edota <b>d'Alembert-Gauss-en teorema</b> ere deitzen denaren arabera, <a href="/wiki/Zenbaki_konplexu" title="Zenbaki konplexu">koefiziente konplexuak</a> dituen <a href="/wiki/Aldagai_(matematika)" title="Aldagai (matematika)">aldagai</a> bakarreko edozein <a href="/wiki/Polinomio" title="Polinomio">polinomio</a> ez-konstantek gutxienez <a href="/w/index.php?title=Erro_(matematika)&action=edit&redlink=1" class="new" title="Erro (matematika) (sortu gabe)">erro</a> bat du. Beste modu batean esanda, definizioz baliokidea da zenbaki konplexuen <a href="/wiki/Gorputz_(matematika)" title="Gorputz (matematika)">gorputza</a> <a href="/w/index.php?title=Aljebraikoki_itxia&action=edit&redlink=1" class="new" title="Aljebraikoki itxia (sortu gabe)">aljebraikoki itxia</a> dela esatea. Bereziki, <a href="/wiki/Zenbaki_errealak,_Cantorren_bidea" title="Zenbaki errealak, Cantorren bidea">zenbaki errealak</a> zenbaki konplexuak direnez, koefiziente errealak dituen aldagai bakarreko edozein polinomio ez-konstantek ere gutxienez erro bat du halabeharrez. </p><p>Teoremaren baliokidea den beste enuntziatu posible bat honakoa da: koefiziente konplexuak dituen aldagai bakarreko eta <i>n</i> <a href="/w/index.php?title=Polinomio_baten_maila&action=edit&redlink=1" class="new" title="Polinomio baten maila (sortu gabe)">mailako</a> edozein polinomio ez-konstantek, <a href="/w/index.php?title=Anizkoiztasuna&action=edit&redlink=1" class="new" title="Anizkoiztasuna (sortu gabe)">anizkoiztasuna</a> kontuan hartuta, zehatz-mehatz <i>n</i> erro konplexu ditu. Bi enuntziatuen arteko baliokidetasuna <a href="/w/index.php?title=Polinomioen_arteko_zatiketaren_algoritmoa&action=edit&redlink=1" class="new" title="Polinomioen arteko zatiketaren algoritmoa (sortu gabe)">polinomioen arteko zatiketaren algoritmoa</a> behin eta berriz aplikatuz frogatu daiteke. </p><p>Nahiz eta teoremaren izenak "aljebraiko" esan, ez dago teorema honen guztiz aljebraikoa den froga, aurrerago ikusiko dugun moduan. Izan ere, aljebraren oinarrizko teorema frogatzeko, gaur egun<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> arte egiaztatu ahal izan den arabera gutxienez, beharrezkoa da <a href="/w/index.php?title=Zenbaki_errealen_osotasuna&action=edit&redlink=1" class="new" title="Zenbaki errealen osotasuna (sortu gabe)">zenbaki errealen osotasun</a> propietatearen forma analitikoren bat erabiltzea eta azken hau ez da kontzeptu aljebraikoa kontsideratzen. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=1" title="Aldatu atal hau: «Historia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=1" title="Edit section's source code: Historia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Petrus_Roth&action=edit&redlink=1" class="new" title="Petrus Roth (sortu gabe)">Petrus Roth</a> izeneko matematikariak, <i>Arithmetica Philosophica</i> (1608) deitutako haren liburuan <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> mailako eta koefiziente errealak dituen <a href="/wiki/Ekuazio" title="Ekuazio">ekuazio</a> polinomiko batek <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> soluzio posible <i>izan ditzakela</i> idatzi zuen. Aldiz, <i>L'invention nouvelle en l'Algebre</i> (1629) liburuan, <a href="/w/index.php?title=Albert_Girard&action=edit&redlink=1" class="new" title="Albert Girard (sortu gabe)">Albert Girard</a> matematikariak <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> mailako ekuazio batek <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> soluzio <i>izan behar dituela</i> ziurtatu zuen, baina ez zuen baieztatzen soluzio hauek zenbaki errealak izan behar direnik. Ez hori bakarrik, baieztapen hau egia dela ziurtatzen du gutxienez <a href="/w/index.php?title=Ekuazio_polinomikoa_osatugabea&action=edit&redlink=1" class="new" title="Ekuazio polinomikoa osatugabea (sortu gabe)">ekuazio polinomikoa osatugabea</a> ez bada. Ekuazio polinomiko bat osatua dela deritzogu haren koefiziente guztiak ez-nuluak baldin badira, bestela osatugabea dela esango dugu. Hala ere, xehetasun hau komentatzerakoan, Albertek benetan uste zuen baieztapena kasu guztietan egia dela, baina ez zuen lortu hau frogatzea. Izan ere, ekuazio polinomiko osatugabe baten adibidea eskeini zuen, hurrengoa hain zuzen ere: <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^{4}=4X-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>X</mi> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{4}=4X-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f7dbf7aa002b7c7ec9c8353cd8f2346b279c0fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.295ex; height:2.843ex;" alt="{\displaystyle X^{4}=4X-3}"></span>. Nahiz eta laugarren mailako ekuazio osatugabea izan, konprobatu daiteke hiru soluzio dituela, <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,-1+i{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,-1+i{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a545048f847d11b61aeb78e2c0dffa23d7c1efc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.908ex; height:3.009ex;" alt="{\displaystyle 1,-1+i{\sqrt {2}}}"></span> eta <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-i{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1-i{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9ce3fc3dcca8876a03bed4b67c51f190cd68f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.712ex; height:3.009ex;" alt="{\displaystyle -1-i{\sqrt {2}}}"></span> non <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span> soluzioak anizkoiztasun bikoitza duen. Beraz, anizkoiztasuna kontuan hartuta 4 soluzio posible ditugu eta honek Albertek egindako baieztapena egia izan zitekeela iradokitzen duen. Bestalde, <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>-ek 1702. urtean kontrakoa susmatzen zuela adierazi zuen eta beranduago <a href="/w/index.php?title=Nikolaus_I_Bernoulli&action=edit&redlink=1" class="new" title="Nikolaus I Bernoulli (sortu gabe)">Nikolaus I Bernoulli</a> matematikari suitzarrak aieru berdina egin zuen. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fitxategi:Leonhard_Euler.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/220px-Leonhard_Euler.jpg" decoding="async" width="220" height="284" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/330px-Leonhard_Euler.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Leonhard_Euler.jpg/440px-Leonhard_Euler.jpg 2x" data-file-width="4672" data-file-height="6040" /></a><figcaption>Leonhard Euler</figcaption></figure> <p>Teorema frogatzea lortu baino lehen, hau ezeztatzeko saiakerak egin ziren. Zehazki, teorema hau egia dela konprobatzen bada, modu sinple batean ondoriozta daiteke koefiziente errealak dituen maila positiboko edozein polinomio, koefiziente errealetako lehenengo eta bigarren mailako polinomioen arteko biderkadura bezala adierazi daitekeela. Hori dela eta, Leibnizek arazo bat zegoela uste zuen: <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> zenbaki erreal ez-nulua bada orduan <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^{4}+a^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{4}+a^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4aa6f0e6c1df268237ec790fe1a2ca7b9936893" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.176ex; height:2.843ex;" alt="{\displaystyle X^{4}+a^{4}}"></span> motako polinomioa ezin da aurretik aipatu dugun moduan adierazi. Modu berean, Bernoullik uste zuen <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^{4}-4X^{3}+2X^{2}+4X+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>X</mi> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{4}-4X^{3}+2X^{2}+4X+4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8df03542de51c0b9db3a8149074bb7c4ed7248b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.145ex; height:2.843ex;" alt="{\displaystyle X^{4}-4X^{3}+2X^{2}+4X+4}"></span> polinomioa ezin zela faktorizatu. Ildo honetatik, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>-ek 1742. urtean gutun bat<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> idatzi zuen eta bai Leibniz eta bai Bernoulli okerrean zeudela adierazi zuen. Izan ere, <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^{4}+a^{4}=(X^{2}+a{\sqrt {2}}X+a^{2})(X^{2}-a{\sqrt {2}}X+a^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>X</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>X</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{4}+a^{4}=(X^{2}+a{\sqrt {2}}X+a^{2})(X^{2}-a{\sqrt {2}}X+a^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0fc9352066be7613cf2ee1a006b82bac3817e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.541ex; height:3.176ex;" alt="{\displaystyle X^{4}+a^{4}=(X^{2}+a{\sqrt {2}}X+a^{2})(X^{2}-a{\sqrt {2}}X+a^{2})}"></span> eta <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^{4}-4X^{3}+2X^{2}+4X+4=(X^{2}-(2+\alpha )X+1+{\sqrt {7}}+\alpha )(X^{2}-(2-\alpha )X+1+{\sqrt {7}}-\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>X</mi> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>X</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>+</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>X</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{4}-4X^{3}+2X^{2}+4X+4=(X^{2}-(2+\alpha )X+1+{\sqrt {7}}+\alpha )(X^{2}-(2-\alpha )X+1+{\sqrt {7}}-\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/260ed1ab3ea216894b69b3fe7e6f0d131171ef7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:92.743ex; height:3.176ex;" alt="{\displaystyle X^{4}-4X^{3}+2X^{2}+4X+4=(X^{2}-(2+\alpha )X+1+{\sqrt {7}}+\alpha )(X^{2}-(2-\alpha )X+1+{\sqrt {7}}-\alpha )}"></span> berdintzak betetzen direla frogatu zuen non <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 =4+2{\sqrt {7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>7</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =4+2{\sqrt {7}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c03e985955f7cf437fcff25b1bbf7db018c750e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.85ex; height:3.009ex;" alt="{\displaystyle \alpha =4+2{\sqrt {7}}}"></span> den. </p><p>Teorema frogatzeko lehenengo saiakera <a href="/wiki/Jean_le_Rond_d%27Alembert" title="Jean le Rond d'Alembert">d'Alembert</a>-ek egin zuen 1746. urtean. Bere frogak akats bat zeukan, momentu horretan frogatuta ez zegoen emaitza bat erabiltzen zuelako, <a href="/w/index.php?title=Puiseuxen_teorema&action=edit&redlink=1" class="new" title="Puiseuxen teorema (sortu gabe)">Puiseux-en teoremaren</a> izenarekin gaur egun ezaguna dena. Azkeneko teorema hau mende bat geroago frogatzea lortu zen eta horrek d'Alemberten froga zuzentzen zuen. </p><p>XVIII. mendearen bukaeran, besbe bi saiakera egin ziren <a href="/w/index.php?title=James_Wood&action=edit&redlink=1" class="new" title="James Wood (sortu gabe)">James Wood</a> eta <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a> matematikarienak alegia, baina bata bestea bezain oker zegoen. Azkenean, 1806. urtean <a href="/w/index.php?title=Jean-Robert_Argand&action=edit&redlink=1" class="new" title="Jean-Robert Argand (sortu gabe)">Jean-Robert Argand</a> matematikariak froga zuzen bat eskaintzea lortu zuen. Teorema enuntziatzerakoan, Aljebraren oinarrizko teorema koefiziente konplexuko polinomioentzat idatzi zuen. Honen ondoren, Gaussek teoremaren beste bi froga zuzen<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><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><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> eman zituen, hauetako bat bere hasierako frogaren bertsio eraldatu bat<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> izanda, honakoan bai guztiz zuzena zena. </p><p>Literaturan Aljebraren oinarrizko teoremaren frogaren lehenego aipamena <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a>-k idatzitako <i>Course d'anlyse de l'École Royale Polytechnique</i><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> (1821) liburuan egiten da. Froga hau Argandena izan arren, Cauchyk ez zuen honen lana aipatzen. </p><p>Azkenik, aipatzekoa da aurreko frogetatik ez dagoela ezta <a href="/w/index.php?title=Froga_eraikitzailea&action=edit&redlink=1" class="new" title="Froga eraikitzailea (sortu gabe)">froga eraikitzaile</a> bat. Teorema honen lehenengo froga eraikitzailea <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Weierstrass</a>-ek eman zuen XIX. mendearen erdialdean eta 1891. urtean hau argitaratu<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> zuen. XX. mendean murgiltzen bagara, <a href="/w/index.php?title=Hellmuth_Knesser&action=edit&redlink=1" class="new" title="Hellmuth Knesser (sortu gabe)">Hellmuth Knesser</a>-ek 1940. urtean horrelako beste froga<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> berri bat gehitzea lortu zuen, gerora 1981. urtean bere semeak, <a href="/w/index.php?title=Martin_Knesser&action=edit&redlink=1" class="new" title="Martin Knesser (sortu gabe)">Martin Knesser</a>-ek sinplifikatu<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> zuena. </p> <div class="mw-heading mw-heading2"><h2 id="Enuntziatu_baliokideak">Enuntziatu baliokideak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=2" title="Aldatu atal hau: «Enuntziatu baliokideak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=2" title="Edit section's source code: Enuntziatu baliokideak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aljebraren oinarrizko teoremak hainbat baliokidetasun ditu: </p> <ul><li>Baldin eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \mathbb {C} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {C} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd49b258febe08f1784bacb8df5173cc37a214f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.071ex; height:2.843ex;" alt="{\displaystyle f\in \mathbb {C} [X]}"></span> eta <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 \deg(f)\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f)\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f4922de55ef45eed15efb73c0290fba3168139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.836ex; height:2.843ex;" alt="{\displaystyle \deg(f)\geq 1}"></span> bada, orduan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> polinomioak gutxienez erro bat du <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> gorputzean.</li> <li>Baldin eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \mathbb {C} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {C} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd49b258febe08f1784bacb8df5173cc37a214f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.071ex; height:2.843ex;" alt="{\displaystyle f\in \mathbb {C} [X]}"></span> eta <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 \deg(f)=n\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f)=n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc35ec576c8e73daece3730164e41dc0210c49e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.329ex; height:2.843ex;" alt="{\displaystyle \deg(f)=n\geq 1}"></span> bada, orduan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> polinomioak zehazki <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> erro ditu <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> gorputzean, erro bakoitza bere anizkoiztasuna beste aldiz kontatuta.</li> <li><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 {C} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6c0e47242946827f33aad1803d66106d2f0307e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [X]}"></span> eraztuneko polinomio irreduzible bakarrak lehenengo mailakoak dira.</li> <li>Baldin eta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \mathbb {C} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in \mathbb {C} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd49b258febe08f1784bacb8df5173cc37a214f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.071ex; height:2.843ex;" alt="{\displaystyle f\in \mathbb {C} [X]}"></span> eta <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 \deg(f)\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \deg(f)\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f4922de55ef45eed15efb73c0290fba3168139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.836ex; height:2.843ex;" alt="{\displaystyle \deg(f)\geq 1}"></span> bada, orduan existitzen dira <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,a_{1},\ldots ,a_{t}\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,a_{1},\ldots ,a_{t}\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d85f8fe1222c2f39633744c9fce0c836130bbf05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.078ex; height:2.509ex;" alt="{\displaystyle c,a_{1},\ldots ,a_{t}\in \mathbb {C} }"></span> eta <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_{1},\ldots ,n_{t}\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1},\ldots ,n_{t}\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90444a129d73d4c756d4797db9ad20554b06da40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.367ex; height:2.509ex;" alt="{\displaystyle n_{1},\ldots ,n_{t}\in \mathbb {N} }"></span> non <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(X)=c(X-a_{1})^{n_{1}}\cdots (X-a_{t})^{n_{t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>c</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(X)=c(X-a_{1})^{n_{1}}\cdots (X-a_{t})^{n_{t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c599f4b54778adde55af4148c56bfd301729ca09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.185ex; height:2.843ex;" alt="{\displaystyle f(X)=c(X-a_{1})^{n_{1}}\cdots (X-a_{t})^{n_{t}}}"></span> den.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Frogak">Frogak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=3" title="Aldatu atal hau: «Frogak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=3" title="Edit section's source code: Frogak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Liouville-ren_teorema_erabilita">Liouville-ren teorema erabilita</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=4" title="Aldatu atal hau: «Liouville-ren teorema erabilita»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=4" title="Edit section's source code: Liouville-ren teorema erabilita"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Izan bedi <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\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}"></span> mailako <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> polinomioa. Jakina da <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> <a href="/wiki/Funtzio_(matematika)" title="Funtzio (matematika)">funtzio</a> <a href="/w/index.php?title=Funtzio_osoa&action=edit&redlink=1" class="new" title="Funtzio osoa (sortu gabe)">osoa</a> dela eta <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> konstante positibo bakoitzarentzat existitzen da <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> zenbaki erreal positiboa non <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 \vert P(z)\vert >m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>P</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mo>></mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert P(z)\vert >m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e24c09a44e8a7a2727bf0ee49682ebe73b2a7e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.075ex; height:2.843ex;" alt="{\displaystyle \vert P(z)\vert >m}"></span> den <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 \vert z\vert >r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>z</mi> <mo fence="false" stretchy="false">|</mo> <mo>></mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert z\vert >r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a190801a52305e8e6ca601fe0ac36d4f6718f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.529ex; height:2.843ex;" alt="{\displaystyle \vert z\vert >r}"></span> denean. <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> polinomioak errorik ez badu, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1/P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=1/P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08224f888a4f2df8963db652511d1cb044cb8287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.447ex; height:2.843ex;" alt="{\displaystyle f=1/P}"></span> moduan definitutako funtzioa osoa da ere eta gainera aurreko esaldian erabilitako propietatea berridatziz, edozein zenbaki positibo erreal <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 \epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3837cad72483d97bcdde49c85d3b7b859fb3fd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }"></span> emanda, <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> zenbaki erreal positibo bat existitzen da non <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 \vert f(z)\vert <\epsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">|</mo> <mo><</mo> <mi>ϵ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert f(z)\vert <\epsilon }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef04e7f33ac3273b0791c16c43ddf98f1b14af2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.512ex; height:2.843ex;" alt="{\displaystyle \vert f(z)\vert <\epsilon }"></span> den <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 \vert z\vert >r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>z</mi> <mo fence="false" stretchy="false">|</mo> <mo>></mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert z\vert >r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a190801a52305e8e6ca601fe0ac36d4f6718f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.529ex; height:2.843ex;" alt="{\displaystyle \vert z\vert >r}"></span> denean. Ondorioz, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> funtzioa <a href="/w/index.php?title=Funtzio_bornatua&action=edit&redlink=1" class="new" title="Funtzio bornatua (sortu gabe)">bornatua</a> da. <a href="/w/index.php?title=Liouvilleren_teorema&action=edit&redlink=1" class="new" title="Liouvilleren teorema (sortu gabe)">Liouvilleren teoremaren</a><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> arabera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> funtzioa osoa eta bornatua denez, halabeharrez <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> konstantea izan behar da eta kontraesan batera ailegatzen gara <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> polinomioaren maila bat delako gutxienez. Horrela, <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> aljebraikoki itxia dela frogatuta gelditzen da. </p><p>Aurreko argudiaketarekin jarraituta, are gehiago frogatu daiteke. Ikusi dugu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> ez dela funtzio osoa eta beraz <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>-k erro bat du gutxienez. Gauzak horrela, <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(z)=(z-\alpha _{1})Q(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(z)=(z-\alpha _{1})Q(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fcc6d55bfd630758c8019a55b58d05eb852cdf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.757ex; height:2.843ex;" alt="{\displaystyle P(z)=(z-\alpha _{1})Q(z)}"></span> idatzi daiteke non <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 _{1}\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1}\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f56abfc6f6771024a2cde42b72d349252e8cfd4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.061ex; height:2.509ex;" alt="{\displaystyle \alpha _{1}\in \mathbb {C} }"></span> balioa <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> polinomioaren erro bat den eta <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> polinomioa <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-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> maila duen. Arrazoiketa analogoa erabiliz, <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> polinomioak erro bat du gutxienez eta berriro faktorizatu daiteke. Prozesu hau <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-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}"></span> aldiz errepikatuta, <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> polinomioa honela idatzi daiteke <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(z)=k(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(z)=k(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e450b2dc4438f38b6b535763d132d8f9f8e1839f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.453ex; height:2.843ex;" alt="{\displaystyle P(z)=k(z-\alpha _{1})(z-\alpha _{2})\cdots (z-\alpha _{n})}"></span> non <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 _{1},\ldots ,\alpha _{n}\in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76653ffe6d91dc6b53e21790e5ae39ea79ca14bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.945ex; height:2.509ex;" alt="{\displaystyle \alpha _{1},\ldots ,\alpha _{n}\in \mathbb {C} }"></span> balioak <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> polinomioaren erroak diren (ez halabeharrez desberdinak). Ohartu <a href="/wiki/Indukzio_matematiko" title="Indukzio matematiko">indukzioaren</a> azken pausuan lehenengo mailako polinomio bat gelditzen zaigula konstante batengatik biderkatuta. </p> <div class="mw-heading mw-heading3"><h3 id="Galois-en_teoria_erabilita">Galois-en teoria erabilita</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=5" title="Aldatu atal hau: «Galois-en teoria erabilita»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=5" title="Edit section's source code: Galois-en teoria erabilita"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Izan bedi <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\in \mathbb {C} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in \mathbb {C} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80edee869b2a535fc343dbc4767b78d9d36e347b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.908ex; height:2.843ex;" alt="{\displaystyle g\in \mathbb {C} [X]}"></span> eta froga dezagun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> polinomioaren erroak zenbaki konplexuen gorputzean daudela. Definitzen dugu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> polinomioa <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> polinomioaren eta bere konjokatuaren arteko biderkadura bezala. Ohartu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> polinomioa <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 {R} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]}"></span> eraztunean aurkitzen dela, hau da, haren koefizienteak zenbaki errealak direla eta gainera <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>-ren erroak <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>-renak ere badira definizioz. Nahikoa da beraz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> polinomioaren erroak <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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>-n daudela frogatzearekin. </p><p>Izan bedi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> <a href="/w/index.php?title=Polinomio_baten_deskonposizio_gorputza&action=edit&redlink=1" class="new" title="Polinomio baten deskonposizio gorputza (sortu gabe)">polinomioaren deskonposizio gorputza</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> gainean. Orduan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24d07e61816d6b60ea81270ea5866f8c70473ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.423ex; height:2.843ex;" alt="{\displaystyle L/\mathbb {C} }"></span> <a href="/w/index.php?title=Gorputz_hedadura&action=edit&redlink=1" class="new" title="Gorputz hedadura (sortu gabe)">gorputz hedadura</a> <a href="/w/index.php?title=Galoisen_hedadura&action=edit&redlink=1" class="new" title="Galoisen hedadura (sortu gabe)">Galoisen hedadura</a><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> da eta ohartu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b8290336ce4054babe270280eb72816b2fb0462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.423ex; height:2.843ex;" alt="{\displaystyle L/\mathbb {R} }"></span> ere orduan Galoisen hedadura bat dela. Kontsidera dezagun azken Galoisen hedadura honen <a href="/wiki/Galoisen_teoria" title="Galoisen teoria">Galoisen taldea</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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> bezala denotatuko duguna. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> taldearen ordena bikoitia denez, existitzen dira <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,m\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,m\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea8f4e6548e9210a21ea55762faeddb45d608d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.823ex; height:2.509ex;" alt="{\displaystyle a,m\in \mathbb {N} }"></span> non <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> zenbaki bakoitia eta <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 \vert G\vert =2^{a+1}\cdot m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>G</mi> <mo fence="false" stretchy="false">|</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert G\vert =2^{a+1}\cdot m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/586a8783bd23c73b8806f716d2d5bc860c245a3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.303ex; height:3.176ex;" alt="{\displaystyle \vert G\vert =2^{a+1}\cdot m}"></span> den. <a href="/w/index.php?title=Sylow-ren_teoremak&action=edit&redlink=1" class="new" title="Sylow-ren teoremak (sortu gabe)">Sylow-ren lehenengo teoremaren</a> arabera, existitzen da <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 H\leq G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>≤</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\leq G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c017471513c8b1345c68cf521c2a7c451bad4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.989ex; height:2.343ex;" alt="{\displaystyle H\leq G}"></span> azpitaldea non haren ordena <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 2^{a+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{a+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74f75adc8f9b6ee6cbc3d4e5f7f4f0fba163c3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.365ex; height:2.676ex;" alt="{\displaystyle 2^{a+1}}"></span> den. Gauzak horrela, <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> azpitaldearen indizea <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>-n <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> dela badakigu. Defini dezagun <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span> gorputza <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> azpitaldeak finko uzten dituen elementuen multzoa bezala, hots, <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={\mathcal {F}}(H)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\mathcal {F}}(H)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74474388f216ab14d8d5af3833677f47fbd0b1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.674ex; height:2.843ex;" alt="{\displaystyle E={\mathcal {F}}(H)}"></span>. Beraz, <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/\mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E/\mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/256ef1f55c6f33694a3f940178f291597a3f093e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.616ex; height:2.843ex;" alt="{\displaystyle E/\mathbb {R} }"></span> <a href="/w/index.php?title=Gorputz_hedaduraren_maila&action=edit&redlink=1" class="new" title="Gorputz hedaduraren maila (sortu gabe)">gorputz hedaduraren maila</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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> bakoitia da. Egoera honetan, zenbaki errealen gorputzak ez duenez maila bakoitiko hedadurarik onartzen, halabeharrez <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 m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"></span> eta ondorioz <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 \vert G\vert =2^{a+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>G</mi> <mo fence="false" stretchy="false">|</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert G\vert =2^{a+1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cc7fa2fe0f540a447282e5581c9d225b918c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.584ex; height:3.176ex;" alt="{\displaystyle \vert G\vert =2^{a+1}}"></span>. </p><p>Bestalde, kontsidera dezagun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24d07e61816d6b60ea81270ea5866f8c70473ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.423ex; height:2.843ex;" alt="{\displaystyle L/\mathbb {C} }"></span> gorputz hedaduraren <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> Galoisen taldea non badakigun <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 \vert T\vert =2^{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>T</mi> <mo fence="false" stretchy="false">|</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert T\vert =2^{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51081c2a9ab56027a6df2b64030f26407554179d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.293ex; height:2.843ex;" alt="{\displaystyle \vert T\vert =2^{a}}"></span> den. Absurdora eramanez, suposa dezagun <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L/\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L/\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24d07e61816d6b60ea81270ea5866f8c70473ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.423ex; height:2.843ex;" alt="{\displaystyle L/\mathbb {C} }"></span> gorputz hedaduraren maila bat baino handiagoa dela. Orduan <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\geq 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06f20f72b40654833aff35ad637c3bb7c36fe5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.491ex; height:2.343ex;" alt="{\displaystyle a\geq 1}"></span> eta ondorioz existitzen da <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 H_{0}\leq T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{0}\leq T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33b91498f6b8e8ec9cf09bfcb21b035adbc56d87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.72ex; height:2.509ex;" alt="{\displaystyle H_{0}\leq T}"></span> azpitaldea non <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 \vert H_{0}\vert =2^{a-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo fence="false" stretchy="false">|</mo> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert H_{0}\vert =2^{a-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64c19a09bf4d503f514a88a1d0ef7e6de9d9fab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.742ex; height:3.176ex;" alt="{\displaystyle \vert H_{0}\vert =2^{a-1}}"></span> den. Azken honen elementu finkoen <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411d268de7b1cf300d7481e3fe59f3b20887e0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}"></span> gorputza bada, <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_{0}\colon \mathbb {C} ]=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [E_{0}\colon \mathbb {C} ]=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29f422413de52f70dd9241d42c05b9367c4187c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.036ex; height:2.843ex;" alt="{\displaystyle [E_{0}\colon \mathbb {C} ]=2}"></span> izan behar da. Baina azken hau ezinezkoa da, zenbaki konplexuen gorputzak ez duelako bigarren mailako hedadurarik. Ondorioz, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974431ad94c29716e7842253365a08ec113bacc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.359ex; height:2.176ex;" alt="{\displaystyle L=\mathbb {C} }"></span> da eta teorema frogatuta gelditzen da. </p> <div class="mw-heading mw-heading2"><h2 id="Korolarioak">Korolarioak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=6" title="Aldatu atal hau: «Korolarioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=6" title="Edit section's source code: Korolarioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aljebraren oinarrizko teorematik emaitza desberdinak ondoriozta daitezke. </p> <ul><li>Zenbaki errealen gorputzaren <a href="/w/index.php?title=Itxitura_aljebraiko&action=edit&redlink=1" class="new" title="Itxitura aljebraiko (sortu gabe)">itxitura aljebraikoa</a> zenbaki konplexuen gorputza da.</li> <li>Zenbaki errealen gorputzaren <a href="/w/index.php?title=Hedadura_aljebraiko&action=edit&redlink=1" class="new" title="Hedadura aljebraiko (sortu gabe)">hedadura aljebraiko</a> posibleak bi dira bakarrik, <a href="/wiki/Isomorfismo" title="Isomorfismo">isomorfismoak</a> gorabehera: zenbaki errealen gorputza bera edo zenbaki konplexuen gorputza.</li> <li>Aldagai bakarreko eta koefiziente arrazionalak dituen edozein <a href="/w/index.php?title=Polinomio_monikoa&action=edit&redlink=1" class="new" title="Polinomio monikoa (sortu gabe)">polinomio moniko</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 X+a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81aec6d1e58c5f94ba1fe339e58239f58f9135f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.05ex; height:2.343ex;" alt="{\displaystyle X+a}"></span> formako binomioen eta <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}+bX+c}"> <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>b</mi> <mi>X</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X^{2}+bX+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/273efd205e97df58c760856f37d7bbc4df1b6b9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.716ex; height:2.843ex;" alt="{\displaystyle X^{2}+bX+c}"></span> formako trinomioen arteko biderkadura da non <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,c\in \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295f2d4962a99e692bb19ff7ca7597a9072e5e22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.951ex; height:2.509ex;" alt="{\displaystyle a,b,c\in \mathbb {Q} }"></span> eta <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 b^{2}-4c<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>c</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}-4c<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92febaee1cbad6d16be42399637c06f506a15795" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.322ex; height:2.843ex;" alt="{\displaystyle b^{2}-4c<0}"></span> den.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Errefentziak">Errefentziak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=7" title="Aldatu atal hau: «Errefentziak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=7" title="Edit section's source code: Errefentziak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation"> <a rel="nofollow" class="external text" href="http://archive.org/details/numbers0000unse_d4i8"><i>Numbers. </i></a> New York : Springer 1995 <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-387-97497-2" title="Berezi:BookSources/978-0-387-97497-2">978-0-387-97497-2</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFEshel2011-12-25">Eshel, Gidon.  (2011-12-25). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.23943/princeton/9780691128917.003.0010">«The Fundamental Theorem of Linear Algebra»</a> <i>Spatiotemporal Data Analysis</i> (Princeton University Press) <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-691-12891-7" title="Berezi:BookSources/978-0-691-12891-7">978-0-691-12891-7</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFWaerdenArtinNoether2003">Waerden, Bartel L. van der; Artin, Emil; Noether, Emmy.  (2003). <i>Algebra. 1. </i> (1. softcover print. argitaraldia) Springer <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-387-40624-4" title="Berezi:BookSources/978-0-387-40624-4">978-0-387-40624-4</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFde_Oliveira2011-03-04">de Oliveira, Oswaldo Rio Branco.  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M.; Stallings, John R..  (1988-05). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307/2047574">«Shorter Notes: On Gauss's First Proof of the Fundamental Theorem of Algebra»</a> <i>Proceedings of the American Mathematical Society</i> 103 (1): 331.  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307%2F2047574">10.2307/2047574</a></span>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0002-9939">0002-9939</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-7"><a href="#cite_ref-7">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Latinez dago aipu honen iturburua"><b>(Latinez)</b></span></span> <span class="citation"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1017/cbo9781139058247.001">«OMNEM FUNCTIONEM ALGEBRAICAM RATIONALEM INTEGRAM UNIUS VARIABILIS»</a> <i>Werke</i> (Cambridge University Press): 3–30. 2011-11-03 <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-8"><a href="#cite_ref-8">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Alemanez dago aipu honen iturburua"><b>(Alemanez)</b></span></span> <span class="citation" id="CITEREFGauss1866">Gauss, Carl Friedrich.  (1866). <a rel="nofollow" class="external text" href="https://books.google.es/books?id=WFxYAAAAYAAJ&redir_esc=y"><i>Analysis. </i></a> Königliche Gesellschaft der Wissenschaften zu Göttingen <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-9"><a href="#cite_ref-9">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation"><a rel="nofollow" class="external text" href="https://www.paultaylor.eu/misc/gauss-web.php">«Gauss's second proof of the fundamental theorem of algebra»</a> <i>www.paultaylor.eu</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-10"><a href="#cite_ref-10">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Frantsesez dago aipu honen iturburua"><b>(Frantsesez)</b></span></span> <span class="citation" id="CITEREFCauchy1821">Cauchy, Augustin-Louis (1789-1857) Auteur du texte.  (1821). <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k29058v"><i>Analyse algébrique ([Reprod. en fac-sim.</i></a><i>) / Augustin-Louis Cauchy. </i>] <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/2-87647-053-5" title="Berezi:BookSources/2-87647-053-5">2-87647-053-5</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-11"><a href="#cite_ref-11">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Alemanez dago aipu honen iturburua"><b>(Alemanez)</b></span></span> <span class="citation"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1017/cbo9781139567886.018">«Neuer Beweis des Satzes, dass jede ganze rationale Function einer Veränderlichen dargestellt werden kann als ein Product aus linearen Functionen derselben Veränderlichen»</a> <i>Mathematische Werke</i> (Cambridge University Press): 251–270. 2013-04-18 <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-12"><a href="#cite_ref-12">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Alemanez dago aipu honen iturburua"><b>(Alemanez)</b></span></span> <span class="citation"> <a rel="nofollow" class="external text" href="https://gdz.sub.uni-goettingen.de/id/PPN266833020_0046"><i>Mathematische Zeitschrift. </i></a> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-13"><a href="#cite_ref-13">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Alemanez dago aipu honen iturburua"><b>(Alemanez)</b></span></span> <span class="citation"> <a rel="nofollow" class="external text" href="https://gdz.sub.uni-goettingen.de/id/PPN266833020_0177"><i>Mathematische Zeitschrift. </i></a> Springer <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-14"><a href="#cite_ref-14">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation" id="CITEREFNovozhilov2019">Novozhilov, Artem.  (2019). <a rel="nofollow" class="external text" href="https://www.ndsu.edu/pubweb/~novozhil/Teaching/452%20Data/12.pdf"><i>Complex Analysis. </i></a> Spring</span>.</span> </li> <li id="cite_note-15"><a href="#cite_ref-15">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/0-387-28917-8_1">«Introduction to Galois Theory»</a> <i>Universitext</i> (Springer New York): 1–6. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-0-387-28725-6" title="Berezi:BookSources/978-0-387-28725-6">978-0-387-28725-6</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> <li id="cite_note-16"><a href="#cite_ref-16">↑</a> <span class="reference-text"><span class="citation"> <span style="cursor: help; font-size: 90%; font-family: bold; color: gray" title="Ingelesez dago aipu honen iturburua"><b>(Ingelesez)</b></span></span> <span class="citation"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.5860/choice.45-0324">«Galois theory for beginners: a historical perspective»</a> <i>Choice Reviews Online</i> 45 (01): 45–0324-45-0324. 2007-09-01  <a href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</a>:<span class="neverexpand"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.5860%2Fchoice.45-0324">10.5860/choice.45-0324</a></span>. <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="http://worldcat.org/issn/0009-4978">0009-4978</a>. <span class="reference-accessdate"><small>(Noiz kontsultatua: 2024-03-12)</small></span></span>.</span> </li> </ol> <div class="mw-heading mw-heading2"><h2 id="Ikus,_gainera"><span id="Ikus.2C_gainera"></span>Ikus, gainera</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=8" title="Aldatu atal hau: «Ikus, gainera»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=8" title="Edit section's source code: Ikus, gainera"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Aljebra" title="Aljebra">Aljebra</a></li> <li><a href="/wiki/Polinomio" title="Polinomio">Polinomio</a></li> <li><a href="/wiki/Galoisen_teoria" title="Galoisen teoria">Galoisen teoria</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&veaction=edit&section=9" title="Aldatu atal hau: «Kanpo estekak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lankide:Unai_Troya/Proba_orria&action=edit&section=9" title="Edit section's source code: Kanpo estekak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div>    </div><script>(RLQ=window.RLQ||[]).push(function(){mw.log.warn("Gadget \"ErrefAurrebista\" was not loaded. 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