代数基本原理

Preface

1 Introduction and Historical Remarks Complex Numbers

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 Eulers Identity

2.5 DeMoivres Theorem for Powers and Roots Exercises

3 Polynomials and Complex Polynomials

3.1 The King 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 Cauchys Theorem

5.1 Line Integrals and Greens Theorem

5.2 Complex Integration and Cauchys Theorem

5.3 The Cauchy Integral Formula and Cauchys Estimate

5.4 Liouviues Theorem and the Fundamental Theorem of Algebra： Proof Two

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 ~r

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 R and Concluding Remarks

Exercises

8 Topology and Topological Spaces

8.1 Winding Number and Proof Five

8.2 Topology——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 Topology and the Final Proof

9.1 Algebraic Topology

9.2 Some Further Group Theory——Abclian 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 Gausss Original Proof

Appendix B： Cauchys 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

《代数基本定理》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。《代数基本定理》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。 《代数基本定理》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《代数基本定理》，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。 《代数基本定理》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。