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?
In order to understand this proof you need to be familar with complex analysis while this result is an algebraic result. Thus, some books feel that algebraic results deserve algebraic proofs. I am familar with another proof that uses Galois theory, however it is little bit more involved, but still nice.
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?