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单词 ProofOfTheFundamentalTheoremOfAlgebraLiouvillesTheorem
释义

proof of the fundamental theorem of algebra (Liouville’s theorem)


Let f: be a polynomialPlanetmathPlanetmath, and suppose f has no root in . We will show f is constant.

Let g=1f. Since f is never zero, g is defined and holomorphic on (ie. it is entire). Moreover, since f is a polynomial, |f(z)| as |z|, and so |g(z)|0 as |z|. Then there is some M>0 such that |g(z)|<1 whenever |z|>M, and g is continuousMathworldPlanetmath and so boundedPlanetmathPlanetmath on the compact set {z:|z|M}.

So g is bounded and entire, and therefore by Liouville’s theorem g is constant. So f is constant as required.

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更新时间:2025/5/4 5:58:11