本书对数学中最重要的定理——代数基本定理给出了六种证明，方法涉及到分析、代数与拓扑等数学分支。全书以一个问题为主线，纵横数学的几乎所有领域，结构严谨、文笔流畅、浅显易懂，适合高年级大学生、研究生自学和讨论，特别适合于用作短学期教材或数学选修类课程教材。

本书对数学中最重要的定理——代数基本定理给出了六种证明，方法涉及到分析、代数与拓扑等数学分支。

本书的六个证明：两个分析方法中一个(本质上)是运用实分析中的两维极值定理，一个是运用标准的复分析方法，也就是经典的Liouville定理；两个代数方法中一个是运用多项式环的知识，一个是运用域扩张的Galois定理；两个拓扑方法中一个是运用分枝数的计算，另一个是运用单位球的基本群。此外附录中给出了Gauss的证明，Cauchy的证明，三个另外的反分析证明以及两个另外的拓扑证明。

本书以一个问题为主线，纵横数学的几乎所有领域，结构严谨、文笔流畅、浅显易懂、引人入胜，是一本少见的能让读者入迷的好读物，可以使读者与作者在书中很好地进行对话与交流。通过学习本书，读者可以增加知识面，加深对学科交叉与渗透的理解和认识。不足之处是各种方法之间缺乏进行比较的描写和分析。

本书适合高年级大学生、研究生自学和讨论，特别适合于用作短学期教材或数学选修类课程教材。

Preface xi

1 Intriduchon and historian hemarto 1

2 Complex Numbers 5

 2.1 Fields and the heal Field 5

 2.2 The Complex Number Field 10

 2.3 Geometrical hepreselltution of Complex Numbers 12

 2.4 POlar FOrm and Euler's ldelltity 14

 2.5 DeMoivre's Theorem for POwers and ROotS 17

 Exercises 19

3 Inlynomds and Complex Anynomall ZI

 3.1 The ac of POlynomials over a Field 21

 3.2 Divisibility and Unique F8ctorhation of Polynomials 24

 3.3 RDotS of POlynomials and F8ctorization 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

 4.2 The Cauchy--Aiemann Equations 41

 4.3 Conformal Mappings and Analyticity 46

 Exercises 49

5 ComDlcx integration and Cauchy's Theorem

 5.1 Line integrals and Green'8 Theorem 52

 5.2 Complex integration and Cauchy's Theorem 61

 5.3 The Cauchv integral Formula and Cauchy's Estimate 66

 5.4 Liouville's Theorem and the Fundamental Theorem of Algebra: Proof ado 70

 5.5 Some Additional insults 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 Permututions and Symmetric Polynomials 86

 6.5 The Fundamental Theorem of Algebra: Proof Three 91

 6.6 An Application--The transcendence of e and ac 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 Alltomornhisms 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 of R and Concluding ramarks 130

 Exercises 132

8 Tbpology and Thpological spaces 134

 8.1 Winding Number and Proof Five 134

 8.2 topology--An Overview 136

 8.3 Continuity and Metric SDaces 138

 8.4 Tbpological Spaces and Homeomorphisms 144

 8.5 Some Further ProDerties of ThDological Spaces 146

 perttes of Thpological Spaces 146

 Exercises 149

9 Algebraic ThpoIOgy and the Final Proof 152

 9.1 Algebraic topology 152

 9.2 Some berther GrouD TheorV--Abelian GrouDs 154

 9.3 Homotopy and the Fundamental Group 159

 9.4 Homology Theory and ThangUlations 166

 9.5 Some Homology Computations 173

 9.6 Homology of Spheres and Brouwer Degree 176

 9.7 The fundamental Theorem of Algebra: Proof Sts 178

 9.8 Concluding ramarks 180

 Exercises 180

Appendly A: A Version of Gau88'8 Original Proof 182

Appendly B: Cauchy'8 Theorem Revisited 187

Appendly C: Three Additions ComPlex A~tic

hoofs of the fundamennd Theorem of Algebra 195

Appendly D: Two More Thpologiod Proofs of the

fundamennd Theorem of Algebra 199

Bibliography and R6ferences 202

Index 2O5