# Complex Polynomials and Minimal Values

moo5003
Hello, I'm trying to prove the fundamental thereom of algebra using the minimal modulous principal. There is one step that I cannot make sense out of logically and I was wondering if one of you could explain it to me.

If the limit as |z| approaches infinity implies |P(z)| approaches infinity for some non-constant polynomial then there exists an a in C such that |P(a)| <or= |P(z)| for all z. Why does this follow?

Homework Helper
Gold Member
Hello, I'm trying to prove the fundamental thereom of algebra using the minimal modulous principal. There is one step that I cannot make sense out of logically and I was wondering if one of you could explain it to me.

If the limit as |z| approaches infinity implies |P(z)| approaches infinity for some non-constant polynomial then there exists an a in C such that |P(a)| <or= |P(z)| for all z. Why does this follow?

I don't quite remember exactly how it goes, but I remember it using 1/|P(z)|.

I'd have to wait until I get home to find out how it went.

moo5003
So, after reading wikipedia for awhile. It follows since you can find a closed disk around 0 with radius r where |f(z)|>or=|f(0)| where |z|>r since the limit goes to infinity. And since the disk is a compact set it must contain a minimum value of z_0. Thus |f(z_0)| <or= |f(x)| for all x in C.

Note: z_0 is on interior of the disk not the boundary.