The fundamental theorem of algebra states that any complex polynomial must have a complex root.
This book examines three pairs of proofs of the theorem from three different areas of
mathematics: abstract algebra complex analysis and topology. The first proof in each pair is
fairly straightforward and depends only on what could be considered elementary mathematics.
However each of these first proofs leads to more general results from which the fundamental
theorem can be deduced as a direct consequence. These general results constitute the second
proof in each pair. To arrive at each of the proofs enough of the general theory of each
relevant area is developed to understand the proof. In addition to the proofs and techniques
themselves many applications such as the insolvability of the quintic and the transcendence of
e and pi are presented. Finally a series of appendices give six additional proofs including a
version of Gauss'original first proof. The book is intended for junior senior level
undergraduate mathematics students or first year graduate students and would make an ideal
capstone course in mathematics.
