抽象代數基礎

內容簡介

《Introduction to Abstract Algebra》(抽象代數基礎)不僅在數學中佔有及其重要的地位，而且在其它學科中也有廣泛的應用，如理論物理、計算機學科等。其研究的方法和觀點，對其他學科產生了越來越大的影響。本教材採取全英文形式撰寫，主要介紹羣、環、域的基本理論。通過《抽象代數》的學習，讓學生理解和掌握羣、環、域三個代數系統的基礎知識和基本理論，受到代數方法的初步訓練，對抽象代數的思想和方法有初步認識，抽象思維能力和邏輯推理能力得到一定提高。從而為進一步學習數學專業其他課程打下必要的基礎。

目錄

Contents

Chapter l Groups and Generating Sets 1

1.1 Binary operations 1

1.2 Isomorphic binary structures 6

Chapter 2 Permutation Groups and Alternating Groups 31

2.1 Permutation groups 31

2.2 Alternating groups 38

Chapter 3 Finitely Generated Abelian Groups and Quotient Groups 45

3.1 The theorem of Lagrange 45

3.2 Finitely generated abelian groups 48

3.3 Properties of homomorphisms 57

3.4 Quotient groups and isomorphism theorems 60

3.5 Automorphism groups 67

Chapter 4 Rings, Quotient Rings and Ideal Theory 78

4.1 Basic definitions 78

4.2 Integral domains 84

4.3 Noncommutative rings 88

4.4 Quatcrnions 95

4.5 Isomorphism thcorcms 101

4.6 Euler’s theorem 107

4.7 Ideal theory 109

Chapter 5 Unique Factorization Domains 119

5.1 Basic definitions 119

5.2 Principal ideal domains 122

5.4 Polynomial rings over UFDs 129

Chapter 6 Extension Fields 141

6.1 Prime fields and extension fields 141

6.2 Algebraic and transcendental elements 145

6.3 Algebraic extensions and algebraic closure 152

6.4 Finite fields 157

Appendix A Equivalence Relations and Quotient Set 165

Appendix B Zorn’s Lemma 167

Appendix C Quotient field 169 ^[3]