[ Home ] [ Up ] [ Info ] [ Mail ]

Prove: Fundamental theorem of algebra. Every polynomial equation

P(z) = a_{0}z^{n} + a_{1}z^{n -1} + ... + a_{n-1}z + a_{n} = 0

of degree n 1 with complex coefficients has at least one root.

Proof. If we assume that P(z) has no root, then the function f(z) = 1/P(z) is analytic for all z (i.e. it possesses no singular points). In addition,

is bounded. Why? Because |f(z)| 0 as |z| . Then by Liouville’s theorem we conclude that f(z) must be a constant. And if f(z) is a constant, so is P(z). But we know that P(z) is not a constant. Thus we have been led to a contradiction and conclude that P(z) = 0 must have at least one root.

[ Home ] [ Up ] [ Info ] [ Mail ]