Why do many books go through "downsizing" arguments to prove the Fundamental Theorem of Algebra? Isn't this the simplest proof?
Theorem. Every non-constant polynomial with complex coefficients has a zero in .
Proof. Let be any polynomial. If for all , is an entire function. Furthermore, if is non-constant, as and is bounded. By Liouville's Theorem, is constant and so , contrary to our assumption.
Yet most books make this so complicated. Why?
What you post is a "complicated" proof: its complications are just hidden in "Liouville's Theorem".
The simplest proof that I know only requires knowing that a complex number can be written in exponential form. Of course it takes quite a lot computation. Is that what you mean by "complicated"?
Theorem. Let . Then has at least one zero.
Proof. Assume to the contrary. Then is a bounded entire function, hence constant by Liouville's Theorem. Contradiction. We have:
Consider two regions in the complex plane. The region inside and on the circle and the region outside the circle . We choose so large such that for all .
So is a bounded entire function and constant.
Why don't Bak and Newman do this?