The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics)

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Product Description

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.

The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) Review

This book will please some and disappoint others, depending on what you seek. Incidentally it supplies 12 proofs of the theorem including Gauss' original proof in his doctoral thesis, half of these proofs being in the appendices.

The authors do a superb job of showing how disparate areas of mathematics can be brought to bear on a single problem from different directions. Many problems would have been suitable for such a purpose but the Fundamental Theorem of Algebra is particularly well chosen due to its importance and the fact that none of the proofs are long.

Yet despite my eagerness to read this book I was disappointed. While utilizing disparate areas of mathematics I wanted more insight into the theorem from almost every one of the proofs. The best proof I have seen of this theorem is in an appendix of G. H. Hardy's book A Course of Pure Mathematics. That proof gives insight as to why the theorem is true because it is a constructive proof (it actually constructs a root) that is easily understood, something not true of all constructive proofs. I was surprised to find Hardy's proof not present in this book in this form but only in a much weaker form (proof five) that is not constructive.

Asking for such clarity of insight is asking too much of most proofs of this theorem but I wanted more in that direction. As a specific example, this theorem can not be proved without appealing somewhere to continuity, which makes the proof itself partake of analysis and not purely algebra. The book does not point out in every proof (e.g., proof four) where such an appeal is made and I suspect it will often be hidden from students who may not grasp that continuity is an essential.

Nevertheless I did enjoy this book and definitely recommend it for the stated purpose of showing how disparate areas of mathematics can be applied to the same problem, for it accomplishes this purpose well. I applaud the authors for a superb effort although, like a literary critic reviewing a play but who can not write one, I wish there had been more. And of course, the authors may read my comments with astonishment and disbelief.

Finally, both mathematical comments in the two-star review by "A Customer" are incorrect; the book is actually correct on these points nor did I find others errors in it.

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