《Springer大學數學圖書·代數基本定理》是2009年11月1日清華大學出版社出版的圖書，作者是(美國)班傑明(Benjamin Fine) (美國)傑哈德(Gerhard Rosenberger)
基本介紹
- 中文名：Springer大學數學圖書·代數基本定理
- 外文名：The Fundamental Theorem of Algebra
- 叢書名：:Springer 大學數學圖書
- 平裝： 208頁
- 正文語種：:英語
圖書信息,作者簡介,內容簡介,目錄,
圖書信息
出版社:清華大學出版社; 第1版 (2009年11月1日)
開本: 16
ISBN: 9787302214793
條形碼: 9787302214793
尺寸: 24 x 17.2 x 1 cm
重量: 340 g
作者簡介
作者：(美國)班傑明(Benjamin Fine) (美國)傑哈德(Gerhard Rosenberger)
內容簡介
《代數基本定理》對數學中最重要的定理——代數基本定理給出了六種證明，方法涉及到分析、代數與拓撲等數學分支。《代數基本定理》的六個證明：兩個分析方法中一個（本質上）是運用實分析中的兩維極值定理，一個是運用標準的複分析方法，也就是經典的Liouville定理；兩個代數方法中一個是運用多項式環的知識，一個是運用域擴張的Galois定理：兩個拓撲方法中一個是運用分枝數的計算，另一個是運用單位球的基本群。此外附錄中給出了Gauss的證明，cauchy的證明，三個另外的反分析證明以及兩個另外的拓撲證明。
《代數基本定理》以一個問題為主線，縱橫數學的幾乎所有領域，結構嚴謹、文筆流暢、淺顯易懂、引人入勝，是一本少見的能讓讀者入迷的好讀物，可以使讀者與作者在書中很好地進行對話與交流。通過學習《代數基本定理》，讀者可以增加知識面，加深對學科交叉與滲透的理解和認識。不足之處是各種方法之間缺乏進行比較的描寫和分析。
《代數基本定理》適合高年級大學生、研究生自學和討論，特別適合於用作短學期教材或數學選修類課程教材。
目錄
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 Euler's Identity
2.5 DeMoivre's 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 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 Liouviue's 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 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 Topological Proofs of the Fundamental Theorem of Algebra
Bibliography and References
Index
