Groups, Rings, and Fields

Contents......Page 6
Preface......Page 8
1.1 Union and Intersection......Page 10
1.2 Venn Diagrams......Page 17
1.3 Mappings......Page 26
1.4 Equivalence Relations......Page 36
1.5 Well-ordering and Induction......Page 43
1.6 Countable Sets......Page 50
2.1 Divisibility......Page 56
2.2 Divisors......Page 59
2.3 Division Algorithm......Page 61
2.4 Euclidean Algorithm......Page 65
2.5 Primes......Page 71
3 Introduction to Rings......Page 80
3.1 Concept of a Polynomial......Page 81
3.2 Division and Euclidean Algorithms......Page 85
3.3 Axioms and Rings......Page 91
4 Introduction to Groups......Page 102
4.1 Semigroups......Page 103
4.2 Finite and Infinite Groups......Page 108
4.3 Lagrange\'s Theorem, Cosets and Conjugacy......Page 136
4.4 Subgroups......Page 124
4.5 Homomorphisms......Page 143
5.1 Arithmetic Modulo n......Page 154
5.2 Integral Domains and Fields......Page 159
5.3 Euclidean Domains......Page 172
5.4 Ideals and Homomorphisms......Page 177
5.5 Principal Ideal and Unique Factorization Domains......Page 186
5.6 Factorization in Q[x]......Page 195
6.1 Permutation Groups......Page 200
6.2 Generators and Relations......Page 213
6.3 Direct Products and Sums......Page 219
6.4 Abelian Groups......Page 222
6.5 p-Groups and Sylow Subgroups......Page 227
Hints to Solutions......Page 234
Suggestions for Further Study......Page 254
Index......Page 255