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?