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Fundamental theorem of algebra - Revision history
2024-03-29T14:56:19Z
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Borishaase at 13:48, 25 July 2022
2022-07-25T13:48:38Z
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<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 13:48, 25 July 2022</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For every non-[[w:Constant function|<span class="wikipedia">constant</span>]] [[w:Polynomial|<span class="wikipedia">polynomial</span>]] <math>p \in \mathbb{C}</math>, there exists some <math>z \in \mathbb{C}</math> such that <math>p(z) = 0</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>For every non-[[w:Constant function|<span class="wikipedia">constant</span>]] [[w:Polynomial|<span class="wikipedia">polynomial</span>]] <math>p \in \mathbb{C}</math>, there exists some <math>z \in \mathbb{C}</math> such that <math>p(z) = 0</math>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''Indirect proof:''' By performing an [[w:Affine transformation|<span class="wikipedia">affine</span>]] [[w:Change of variables|<span class="wikipedia">substitution</span>]] of variables, reduce to the case <math>1/p(0) \ne \mathcal{O}(\<del class="diffchange diffchange-inline">text{d0}</del>)</math>. Suppose that <math>p(z) \ne 0</math> for all <math>z \in \mathbb{C}</math>. Since <math>f(z) := 1/p(z)</math> is [[w:Holomorphic function|<span class="wikipedia">holomorphic</span>]], it holds that <math>f(<del class="diffchange diffchange-inline">1/</del>\<del class="diffchange diffchange-inline">text</del>{<del class="diffchange diffchange-inline">d0</del>}) = \mathcal{O}(\<del class="diffchange diffchange-inline">text{d0}</del>)</math>. By the mean value inequality<ref name="Ribenboim">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3rd, impr. Ed.; 1992; Springer; Berlin; ISBN 9783540552338, p. 160.</ref> <math>|f(0)| \le {|f|}_{\gamma}</math> for <math>\gamma = \partial\mathbb{B}_{r}(0)</math> and arbitrary <math>r \in \mathbb{R}_{&gt;0}</math>, and hence <math>f(0) = \mathcal{O}(\<del class="diffchange diffchange-inline">text{d0}</del>)</math>, which is a contradiction.<math>\square</math></div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''Indirect proof:''' By performing an [[w:Affine transformation|<span class="wikipedia">affine</span>]] [[w:Change of variables|<span class="wikipedia">substitution</span>]] of variables, reduce to the case <math>1/p(0) \ne \mathcal{O}(\<ins class="diffchange diffchange-inline">iota</ins>)</math>. Suppose that <math>p(z) \ne 0</math> for all <math>z \in \mathbb{C}</math>. Since <math>f(z) := 1/p(z)</math> is [[w:Holomorphic function|<span class="wikipedia">holomorphic</span>]], it holds that <math>f(\<ins class="diffchange diffchange-inline">tilde</ins>{<ins class="diffchange diffchange-inline">\iota</ins>}) = \mathcal{O}(\<ins class="diffchange diffchange-inline">iota</ins>)</math>. By the mean value inequality<ref name="Ribenboim">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3rd, impr. Ed.; 1992; Springer; Berlin; ISBN 9783540552338, p. 160.</ref> <math>|f(0)| \le {|f|}_{\gamma}</math> for <math>\gamma = \partial\mathbb{B}_{r}(0)</math> and arbitrary <math>r \in \mathbb{R}_{&gt;0}</math>, and hence <math>f(0) = \mathcal{O}(\<ins class="diffchange diffchange-inline">iota</ins>)</math>, which is a contradiction.<math>\square</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== See also ==</div></td></tr>
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Borishaase
https://en.mwiki.de/index.php?title=Fundamental_theorem_of_algebra&diff=134&oldid=prev
Borishaase: Fundamental theorem of algebra
2020-04-27T22:47:28Z
<p>Fundamental theorem of algebra</p>
<p><b>New page</b></p><div>For every non-[[w:Constant function|<span class="wikipedia">constant</span>]] [[w:Polynomial|<span class="wikipedia">polynomial</span>]] <math>p \in \mathbb{C}</math>, there exists some <math>z \in \mathbb{C}</math> such that <math>p(z) = 0</math>.<br />
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'''Indirect proof:''' By performing an [[w:Affine transformation|<span class="wikipedia">affine</span>]] [[w:Change of variables|<span class="wikipedia">substitution</span>]] of variables, reduce to the case <math>1/p(0) \ne \mathcal{O}(\text{d0})</math>. Suppose that <math>p(z) \ne 0</math> for all <math>z \in \mathbb{C}</math>. Since <math>f(z) := 1/p(z)</math> is [[w:Holomorphic function|<span class="wikipedia">holomorphic</span>]], it holds that <math>f(1/\text{d0}) = \mathcal{O}(\text{d0})</math>. By the mean value inequality<ref name="Ribenboim">[[w:Reinhold Remmert|<span class="wikipedia">Remmert, Reinhold</span>]]: ''Funktionentheorie 1'' : 3rd, impr. Ed.; 1992; Springer; Berlin; ISBN 9783540552338, p. 160.</ref> <math>|f(0)| \le {|f|}_{\gamma}</math> for <math>\gamma = \partial\mathbb{B}_{r}(0)</math> and arbitrary <math>r \in \mathbb{R}_{&gt;0}</math>, and hence <math>f(0) = \mathcal{O}(\text{d0})</math>, which is a contradiction.<math>\square</math><br />
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== See also ==<br />
* [[List of mathematical symbols]]<br />
* [[w:Fundamental theorem of algebra|<span class="wikipedia">Fundamental theorem of algebra</span>]]<br />
[[Category:Areas of mathematics]]<br />
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== Reference ==<br />
<references /><br />
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[[de:Fundamentalsatz der Algebra]]</div>
Borishaase