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代数基本定理 英文PDF|Epub|txt|kindle电子书版本网盘下载
- BenjaminFine，GerhardRosenberger编著 著
- 出版社： 北京：清华大学出版社
- ISBN：9787302214793
- 出版时间：2009
- 标注页数：210页
- 文件大小：8MB
- 文件页数：225页
- 主题词：代数－高等学校－教材－英文
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图书目录
1 Introduction and Historical Remarks1
2 Complex Numbers5
2.1 Fields and the Real Field5
2.2 The Complex Number Field10
2.3 Geometrical Representation of Complex Numbers12
2.4 Polar Form and Euler's Identity14
2.5 DeMoivre's Theorem for Powers and Roots17
Exercises19
3 Polynomials and Complex Polynomials21
3.1 The Ring of Polynomials over a Field21
3.2 Divisibility and Unique Factorization of Polynomials24
3.3 Roots of Polynomials and Factorization27
3.4 Real and Complex Polynomials29
3.5 The Fundamental Theorem of Algebra:Proof One31
3.6 Some Consequences of the Fundamental Theorem33
Exercises34
4 Complex Analysis and Analytic Functions36
4.1 Complex Functions and Analyticity36
4.2 The Cauchy-Riemann Equations41
4.3 Conformal Mappings and Analyticity46
Exercises49
5 Complex Integration and Cauchy's Theorem52
5.1 Line Integrals and Green's Theorem52
5.2 Complex Integration and Cauchy's Theorem61
5.3 The Cauchy Integral Formula and Cauchy's Estimate66
5.4 Liouville's Theorem and the Fundamental Theorem of Algebra:Proof Two70
5.5 Some Additional Results71
5.6 Concluding Remarks on Complex Analysis72
Exercises72
6 Fields and Field Extensions74
6.1 Algebraic Field Extensions74
6.2 Adjoining Roots to Fields81
6.3 Splitting Fields84
6.4 Permutations and Symmetric Polynomials86
6.5 The Fundamental Theorem of Algebra:Proof Three91
6.6 An Application—The Transcendence of e and π94
6.7 The Fundamental Theorem of Symmetric Polynomials99
Exercises102
7 Galois Theory104
7.1 Galois Theory Overview104
7.2 Some Results From Finite Group Theory105
7.3 Galois Extensions112
7.4 Automorphisms and the Galois Group115
7.5 The Fundamental Theorem of Galois Theory119
7.6 The Fundamental Theorem of Algebra:Proof Four123
7.7 Some Additional Applications of Galois Theory124
7.8 Algebraic Extensions of R and Concluding Remarks130
Exercises132
8 Topology and Topological Spaces134
8.1 Winding Number and Proof Five134
8.2 Topology—An Overview136
8.3 Continuity and Metric Spaces138
8.4 Topological Spaces and Homeomorphisms144
8.5 Some Further Properties of Topological Spaces146
Exercises149
9 Algebraic Topology and the Final Proof152
9.1 Algebraic Topology152
9.2 Some Further Group Theory—Abelian Groups154
9.3 Homotopy and the Fundamental Group159
9.4 Homology Theory and Triangulations166
9.5 Some Homology Computations173
9.6 Homology of Spheres and Brouwer Degree176
9.7 The Fundamental Theorem of Algebra:Proof Six178
9.8 Concluding Remarks180
Exercises180
Appendix A:A Version of Gauss's Original Proof182
Appendix B:Cauchy's Theorem Revisited187
Appendix C:Three Additional Complex Analytic Proofs of the Fundamental Theorem of Algebra195
Appendix D:Two More Topological Proofs of the Fundamental Theorem of Algebra199
Bibliography and References202
Index205