My book on Abstract Algebra (John Fragleigh 7th Ed.) is about to prove that every field has a extension field which has algebraic closure. Before it shows that it proves it for complex numbers. More, famously known as the Fundamental Theorem of Algebra.
Let be a non-constant polynomial, in sake of a contradicton assume, that has no zero. Then is analytic everywhere. Then, . Thus, is bounded (this I do not understand). Then by Liouville's Theorem it must be a constant, thus must be a constant thus a contradiction.
Note it is possible that I erred because I am writing this by memory. May someone help explain this proof to me, I am dieing to know the proof to this theorem.
One more question, why is it so fundamental? There are other theorems from algebra which probably have more importance, like that every field has an extension field having algebraic closure, that is more fundamental!