The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) [Hardcover]

Benjamin Fine (Author), Gerhard Rosenberger (Author)

Book Description

Publication Date: June 20, 1997 | ISBN-10: 0387946578 | ISBN-13: 978-0387946573 | Edition: 1

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.

Editorial Reviews

Review

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

Product Details

• Hardcover: 240 pages
• Publisher: Springer; 1 edition (June 20, 1997)
• Language: English
• ISBN-10: 0387946578
• ISBN-13: 978-0387946573
• Product Dimensions: 9.5 x 6.4 x 0.6 inches
• Shipping Weight: 1.1 pounds (View shipping rates and policies)
• Average Customer Review: 3.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #1,825,182 in Books (See Top 100 in Books)

Customer Reviews

Most Helpful Customer Reviews

4 of 4 people found the following review helpful:

4.0 out of 5 stars Will please some and disappoint others, July 19, 2008

By 

rjohnp "rjohnp" (Beaverton, Oregon United States) - See all my reviews

This review is from: The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) (Hardcover)

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.

5 of 7 people found the following review helpful:

5.0 out of 5 stars A wonderful text!, June 16, 2000

By A Customer

This review is from: The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) (Hardcover)

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.

7 of 13 people found the following review helpful:

2.0 out of 5 stars Too Poorly Written, November 13, 2003

By A Customer

This review is from: The Fundamental Theorem of Algebra (Undergraduate Texts in Mathematics) (Hardcover)

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.