Difference between revisions of "Fundamental theorem of algebra"
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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>. | 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>. | ||
| − | '''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}(\ | + | '''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}(\iota)</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(\tilde{\iota}) = \mathcal{O}(\iota)</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}_{>0}</math>, and hence <math>f(0) = \mathcal{O}(\iota)</math>, which is a contradiction.<math>\square</math> |
== See also == | == See also == | ||
Latest revision as of 15:48, 25 July 2022
For every non-constant polynomial [math]\displaystyle{ p \in \mathbb{C} }[/math], there exists some [math]\displaystyle{ z \in \mathbb{C} }[/math] such that [math]\displaystyle{ p(z) = 0 }[/math].
Indirect proof: By performing an affine substitution of variables, reduce to the case [math]\displaystyle{ 1/p(0) \ne \mathcal{O}(\iota) }[/math]. Suppose that [math]\displaystyle{ p(z) \ne 0 }[/math] for all [math]\displaystyle{ z \in \mathbb{C} }[/math]. Since [math]\displaystyle{ f(z) := 1/p(z) }[/math] is holomorphic, it holds that [math]\displaystyle{ f(\tilde{\iota}) = \mathcal{O}(\iota) }[/math]. By the mean value inequality[1] [math]\displaystyle{ |f(0)| \le {|f|}_{\gamma} }[/math] for [math]\displaystyle{ \gamma = \partial\mathbb{B}_{r}(0) }[/math] and arbitrary [math]\displaystyle{ r \in \mathbb{R}_{>0} }[/math], and hence [math]\displaystyle{ f(0) = \mathcal{O}(\iota) }[/math], which is a contradiction.[math]\displaystyle{ \square }[/math]
See also
Reference
- ↑ Remmert, Reinhold: Funktionentheorie 1 : 3rd, impr. Ed.; 1992; Springer; Berlin; ISBN 9783540552338, p. 160.