ProofOfTheFundamentalTheoremOfAlgebra 2002-02-13 09:19:14 2004-09-16 09:23:32 Proof fundamental theorem of algebra 0 % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Let $f \colon \mathbb{C}\rightarrow\mathbb{C}$ be a polynomial, and suppose $f$ has no root in $\mathbb{C}$. We will show $f$ is constant. Let $g=\frac{1}{f}$. Since $f$ is never zero, $g$ is defined and holomorphic on $\mathbb{C}$ (ie. it is entire). Moreover, since $f$ is a polynomial, $|f(z)|\rightarrow\infty$ as $|z|\rightarrow\infty$, and so $|g(z)|\rightarrow 0$ as $|z|\rightarrow\infty$. Then there is some $M>0$ such that $|g(z)|<1$ whenever $|z|>M$, and $g$ is continuous and so bounded on the compact set $\{ z\in\mathbb{C}:|z|\leq M\}$. So $g$ is bounded and entire, and therefore by Liouville's theorem $g$ is constant. So $f$ is constant as required.$\square$