导语

  

内容提要

  

    《抽象代数基础》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。

Chapter 1 Groups and Generating Sets
1.1 Binary operations
1.2 Isomorphic binary structures
1.3 Groups
1.4 Subgroups
1.5 Cyclic groups
1.6 Generating sets
1.7 Exercises
Chapter 2 Permutation Groups and Alternating
Groups
2.1 Permutation groups
2.2 Alternating groups
2.3 Exercises
Chapter 3 Finitely Generated Abelian Groups and
Quotient Groups
3.1 The theorem of Lagrange
3.2 Finitely generated abelian groups
3.3 Properties of homomorphisms
3.4 Quotient groups and isomorphism theorems
3.5 Automorphism groups
3.6 Simple groups
3.7 Exercises
Chapter 4 Rings, Quotient Rings and Ideal Theory
4.1 Basic definitions
4.2 Integral domains
4.3 Noncommutative rings
4.4 Quaternions
4.5 Isomorphism theorems
4.6 Euler's theorem
4.7 Ideal theory
4.8 Exercises
Chapter 5 Unique Factorization Domains
5.1 Basic definitions
5.2 Principal ideal domains
5.3 Euclidean domains
5.4 Polynomial rings over UFDs
5.5 Multiplicative norms
5.6 Exercises
Chapter 6 Extension Fields
6.1 Prime fields and extension fields
6.2 Algebraic and transcendental elements
6.3 Algebraic extensions and algebraic closure
6.4 Finite fields
6.5 Exercises
Appendix A Equivalence Relations and Quotient Set
Appendix B Zorn's Lemma
Appendix C Quotient field
Reference
Index