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topics:the_fundamental_theorem_of_algebra [2019/07/31 07:18] romik created |
topics:the_fundamental_theorem_of_algebra [2019/08/09 16:52] (current) |
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| - | Let $f(z)$ be a polynomial. Assume by contradiction that $f(z)$ does not have a zero. Define a function $g(z) = \frac{1}{f(z)}$. Then $g(z)$ is an entire function, and satisfies $\lim_{z\to\infty} g(z) = 0$. Since it is also bounded in any compact set, it follows that $g(z)$ is a bounded entire function. By Liouville's theorem, $g(z)$ is constant, and therefore $f(z)$ is a constant polynomial. | + | Let $f(z)$ be a polynomial. Assume by contradiction that $f(z)$ does not have a zero. Define a function $g(z) = \frac{1}{f(z)}$. Then $g(z)$ is an entire function, and satisfies $\lim_{z\to\infty} g(z) = 0$. Since it is also bounded in any compact set, it follows that $g(z)$ is a bounded entire function. By [[Liouville's theorem]], $g(z)$ is constant, and therefore $f(z)$ is a constant polynomial. |
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