Customer Reviews

The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics)

5 star:		(1)
4 star:		(1)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

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.

This text gives a number of proofs of the Fundamental Theorem of Algebra (FTA). The proofs come from diverse areas of mathematics including complex analysis, abstract algebra (Galois theory) and topology. A version of Gauss's original proof is even given. The background information leading up to each proof makes the book readable for a math major who has taken a first course in abstract algebra. A student who uses this text will have seen a lot of really interesting mathematics by the end and will have a sense of how the mathematics can be used.

To be sure, the aim of this book is interesting, but it's too poorly written. There are so many serious mistakes that I wonder why the students who took the course of these authors didn't point them out.
For example, the latter half of Lemma 4.1.5 is wrong for it lacks (-i), and Theorem 7.3.3 is wrong for |K:F| is not equal to the number of automorphisms of K fixing F, but is equal to the number of injective isomorphisms form K to algebraic closure of F fixing F.
I recommend you to read more reliable books.