The Fundamental Theorem of Algebra

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August 1997



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 this 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 prooof 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 trascendence 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. It is ideal for a "capstone" course in mathematics. It could also be used as an alternative approach to an undergraduate abstract algebra course. Finally, because of the breadth of topics it covers it would also be ideal for a graduate course for mathmatics teachers.


1 Introduction and Historical Remarks.- 2 Complex Numbers.- 2.1 Fields and the Real Field.- 2.2 The Complex Number Field.- 2.3 Geometrical Representation of Complex Numbers.- 2.4 Polar Form and Euler's Identity.- 2.5 DeMoivre's Theorem for Powers and Roots.- Exercises.- 3 Polynomials and Complex Polynomials.- 3.1 The Ring of Polynomials over a Field.- 3.2 Divisibility and Unique Factorization of Polynomials.- 3.3 Roots of Polynomials and Factorization.- 3.4 Real and Complex Polynomials.- 3.5 The Fundamental Theorem of Algebra: Proof One.- 3.6 Some Consequences of the Fundamental Theorem.- Exercises.- 4 Complex Analysis and Analytic Functions.- 4.1 Complex Functions and Analyticity.- 4.2 The Cauchy-Riemann Equations.- 4.3 Conformal Mappings and Analyticity.- Exercises.- 5 Complex Integration and Cauchy's Theorem.- 5.1 Line Integrals and Green's Theorem.- 5.2 Complex Integration and Cauchy's Theorem.- 5.3 The Cauchy Integral Formula and Cauchy's Estimate.- 5.4 Liouville's Theorem and the Fundamental Theorem of Algebra: Proof Ttvo.- 5.5 Some Additional Results.- 5.6 Concluding Remarks on Complex Analysis.- Exercises.- 6 Fields and Field Extensions.- 6.1 Algebraic Field Extensions.- 6.2 Adjoining Roots to Fields.- 6.3 Splitting Fields.- 6.4 Permutations and Symmetric Polynomials.- 6.5 The Fundamental Theorem of Algebra: Proof Three.- 6.6 An Application-The Transcendence of e and ?.- 6.7 The Fundamental Theorem of Symmetric Polynomials.- Exercises.- 7 Galois Theory.- 7.1 Galois Theory Overview.- 7.2 Some Results From Finite Group Theory.- 7.3 Galois Extensions.- 7.4 Automorphisms and the Galois Group.- 7.5 The Fundamental Theorem of Galois Theory.- 7.6 The Fundamental Theorem of Algebra: Proof Four.- 7.7 Some Additional Applications of Galois Theory.- 7.8 Algebraic Extensions of ? and Concluding Remarks.- Exercises.- 8 7bpology and Topological Spaces.- 8.1 Winding Number and Proof Five.- 8.2 Tbpology-An Overview.- 8.3 Continuity and Metric Spaces.- 8.4 Topological Spaces and Homeomorphisms.- 8.5 Some Further Properties of Topological Spaces.- Exercises.- 9 Algebraic Zbpology and the Final Proof.- 9.1 Algebraic lbpology.- 9.2 Some Further Group Theory-Abelian Groups.- 9.3 Homotopy and the Fundamental Group.- 9.4 Homology Theory and Triangulations.- 9.5 Some Homology Computations.- 9.6 Homology of Spheres and Brouwer Degree.- 9.7 The Fundamental Theorem of Algebra: Proof Six.- 9.8 Concluding Remarks.- Exercises.- Appendix A: A Version of Gauss's 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 Ibpological Proofs of the Fundamental Theorem of Algebra.- Bibliography and References.
EAN: 9780387946573
ISBN: 0387946578
Untertitel: 'Undergraduate Texts in Mathematics'. Mit Abbildungen.
Verlag: Springer-Verlag GmbH
Erscheinungsdatum: August 1997
Seitenanzahl: XI
Format: Pappe
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