Theorem (Fundamental Theorem of Algebra) Every polynomial of degree n \ge 1  with complex coefficients has a zero in C.

Proof

Let p(z) = z^n + a_{n-1}z^{n-1} + \ldots + a_1z + a_0  be a polynomial of degree n \ge 1  and assume that p(z) \ne 0  for all z \in  C.

By Cauchy's integral theorem we have

\oint_{|z| = r} \frac {dz}{zp(z)} = \frac {2\pi i}{p(0)} \ne 0

where the circle is traversed counterclockwise. On the other hand,

| \oint_{|z| = r} \frac{dz}{zp(z)}| \le 2\pi r \times max_{|z| = r} \frac{1}{|zp(z)|} = \frac{2\pi}{min_{|z| = r}|p(z)|} \rightarrow 0

as r \rightarrow \infty  (since |p(z)| \ge |z|^n|1 - \frac{|a_{n-1}|}{|z|} - \ldots - \frac{|a_0|}{|z|^n}| ),  which is a contradiction.

Reference

  • A. R. Schep, A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra, Am Math Monthly 116, Jan 2009, 67-68.