The fundamental theorem of algebra

Benjamin Fine, Gerhard Rosenberger9780387946573, 0-387-94657-8

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 this 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. 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 [rho] and [pi] are presented. Finally, a series of appendices give six additional proofs including a version of Gauss’s original first proof. The book is intended for junior/senior-level undergraduate mathematics students or first-year graduate students. It is ideal for a “capstone” course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathematics teachers.