The Fundamental Theorem of Algebra

The purpose of this book is to examine three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. It is ideal for a "capstone" course in mathematics for junior/senior level undergraduate mathematics students or first year graduate students. It could also be used as an alternative approach to an undergraduate abstract algebra course.

The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies really at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. The purpose of his book is to examine 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 lends itself to generalizations, which in turn, lead 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. In addition to the proofs and techniques themselves, many applications such as the insolvability of the quintic and the trascendence of e and pi are presented. The book is intended for junior/senior level undergraduate mathematics students or first year graduate students. -- Book Description

The authors intend their book as the main text for a senior level "capstone" course, but it could serve as supplementary reading for a variety of other courses, or as a reference. No other book covers similar ground. Highly recommended. -- D.V. Feldman, University of New Hampshire, CHOICE, March 1998