《Introduction to Abstract Algebra》(抽象代数基础)不仅在数学中占有及其重要的地位,而且在其它学科中也有广泛的应用,如理论物理、计算机学科等。其研究的方法和观点,对其他学科产生了越来越大的影响。本教材采取全英文形式撰写,主要介绍群、环、域的基本理论。通过《抽象代数》的学习,让学生理解和掌握群、环、域三个代数系统的基础知识和基本理论,受到代数方法的初步训练,对抽象代数的思想和方法有初步认识,抽象思维能力和逻辑推理能力得到一定提高。从而为进一步学习数学专业其他课程打下必要的基础。
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目录
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