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

Fine, Benjamin; Rosenberger, Gerhard

Hardcover

ISBN 10: 0387946578 ISBN 13: 9780387946573

Publisher: Springer, 1997

Synopsis

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.

About this title

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

About this edition

• Publisher: Springer
• Publication date: 1997
• ISBN 10: 0387946578
• ISBN 13: 9780387946573
• Binding: Hardcover
• Number of pages: 221