ABSTRACT
In Chapter 2 we proved the Fundamental Theorem of Algebra using some basic point-set topology and simple estimates. We now present an almost completely algebraic proof. The main property of the real numbers R that we require is that it is an ordered field. We develop a far-reaching generalisation of Cauchy's Theorem in group theory due to Ludwig Sylow, about the existence of certain subgroups of prime power order in any finite group. We combine Sylow's Theorem with the Galois correspondence to prove the Fundamental Theorem of Algebra in the general context of an algebraically closed field.
