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The Fundamental Theorem of Algebra(Benjamin Fine Gerhard Rosenberger)
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Preface vii 1 Introduction and Historical Remarks 1 2 Complex Numbers 5 2.1 Fields and the Real Field 5 2.2 The Complex Number Field 10 2.3 Geometrical Representation of Complex Numbers 12 2.4 Polar Form and Euler's Identity 14 2.5 DeMoivre's Theorem for Powers and Roots 17 Exercises 19 3 Polynomials and Complex Polynomials 21 3.1 The Ring of Polynomials over a Field 21 3.2 Divisibility and Unique Factorization of Polynomials 24 3.3 Roots of Polynomials and Factorization 27 3.4 Real and Complex Polynomials 29 3.5 The Fundamental Theorem of Algebra: Proof One 31 3.6 Some Consequences of the Fundamental Theorem 33 Exercises 34 4 Complex Analysis and Analytic Functions 36 4.1 Complex Functions and Analyticity 36 Contents Contents xi 9 Algebraic Thpology and the Final Proof 152 9.1 Algebraic Thpology 152 9.2 Some Further Group Theory-Abelian Groups 154 9.3 Homotopy and the Fundamental Group 159 9.4 Homology Theory and 'lliangulations 166 9.5 Some Homology Computations 173 9.6 Homology of Spheres and Brouwer Degree 176 9.7 The Fundamental Theorem of Algebra: Proof Six 178 9.8 Concluding Remarks 180 Exercises 180 Appendix A: A Version of Gauss's Original Proof 182 Appendix B: Cauchy's Theorem Revisited 187 Appendix C: Three Additional Complex Analytic Proofs of the Fundamental Theorem of Algebra 195 Appendix D: Two More Thpologica1 Proofs of the Fundamental Theorem of Algebra 199 Bibliography and References 202 Index 205 4.2 The Cauchy-Riemann Equations 41 4.3 Conformal Mappings and Analyticity 46 Exercises 49 5 Complex Integration and Cauchy's Theorem 52 5.1 Line Integrals and Green's Theorem 52 5.2 Complex Integration and Cauchy'S Theorem 61 5.3 The Cauchy Integral Formula and Cauchy'S Estimate 66 5.4 Liouville's Theorem and the Fundamental Theorem of Algebra: proofTwo 70 5.5 Some Additional Results 71 5.6 Concluding Remarks on Complex Analysis 72 Exercises 72 6 Fields and Field Extensions 74 6.1 Algebraic Field Extensions 74 6.2 Adjoining Roots to Fields 81 6.3 Splitting Fields 84 6.4 Permutations and Symmetric Polynomials 86 6.5 The Fundamental Theorem of Algebra: Proof Three 91 6.6 An Application-The Transcendence of e and rr 94 6.7 The Fundamental Theorem of Symmetric Polynomials 99 Exercises 102 7 Galois Theory 104 7.1 Galois Theory Overview 104 7.2 Some Results From Finite Group Theory 105 7.3 Galois Extensions 112 7.4 Automorphisms and the Galois Group 115 7.5 The Fundamental Theorem of Galois Theory 119 7.6 The Fundamental Theorem of Algebra: Proof Four 123 7.7 Some Additional Applications of Galois Theory 124 7.8 Algebraic Extensions ofR and Concluding Remarks 130 Exercises 132 8 Thpology and Thpological Spaces 134 8.1 Winding Number and proof Five 134 8.2 Thpology-An Overview 136 8.3 Continuity and Metric Spaces 138 8.4 Thpological Spaces and Homeomorphisms 144 8.5 Some Further Properties of Thpological Spaces 146 Exercises |
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