The Fundamental Theorem of Algebra

Hardcover | June 20, 1997

byBenjamin Fine, Gerhard Rosenberger

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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.

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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 cons...

Format:HardcoverDimensions:221 pages, 9.25 × 6.1 × 0 inPublished:June 20, 1997Publisher:Springer New York

The following ISBNs are associated with this title:

ISBN - 10:0387946578

ISBN - 13:9780387946573

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Table of Contents

Contents: Introduction and Historical Remarks.- Complex Numbers.- Polynomials and Complex Polynomials..- Complex Analysis and Analytic Functions..- Complex Integration and Cauchy's Theorem.- Fields and Field Extensions.- Galois Theory.- Topology and Topological Spaces.- Algebraic Topology and the Final Proof.- Appendix A: A Version of Gauss' Original Proof.- Appendix B: Cauchy's Theorem Revisited.- Appendix C: Three Additional Complex Analytic Proofs of the Fundamental Theorem of Algebra.- Appendix D: Two More Topological Proofs of the Fundamental Theorem of Algebra.- Bibliography and References.- Index.