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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