0 Preliminaries 1
0.1 Proofs 1
0.2 Sets 2
0.3 Mappings 3
0.4 Equivalences 4
1 Integers and Permutations 6
1.1 Induction 6
1.2 Divisors and Prime Factorization 8
1.3 Integers Modulo 11
1.4 Permutations 13
2 Groups 17
2.1 Binary Operations 17
2.2 Groups 19
2.3 Subgroups 21
2.4 Cyclic Groups and the Order of an Element 24
2.5 Homomorphisms and Isomorphisms 28
2.6 Cosets and Lagrange's Theorem 30
2.7 Groups of Motions and Symmetries 32
2.8 Normal Subgroups 34
2.9 Factor Groups 36
2.10 The Isomorphism Theorem 38
2.11 An Application to Binary Linear Codes 43
3 Rings 47
3.1 Examples and Basic Properties 47
3.2 Integral Domains and Fields 52
3.3 Ideals and Factor Rings 55
3.4 Homomorphisms 59
3.5 Ordered Integral Domains 62
4 Polynomials 64
4.1 Polynomials 64
4.2 Factorization of Polynomials over a Field 67
4.3 Factor Rings of Polynomials over a Field 70
4.4 Partial Fractions 76
4.5 Symmetric Polynomials 76
5 Factorization in Integral Domains 81
5.1 Irreducibles and Unique Factorization 81
5.2 Principal Ideal Domains 84
6 Fields 88
6.1 Vector Spaces 88
6.2 Algebraic Extensions 90
6.3 Splitting Fields 94
6.4 Finite Fields 96
6.5 Geometric Constructions 98
6.7 An Application to Cyclic and BCH Codes 99
7 Modules over Principal Ideal Domains 102
7.1 Modules 102
7.2 Modules over a Principal Ideal Domain 105
8 p-Groups and the Sylow Theorems 108
8.1 Products and Factors 108
8.2 Cauchy's Theorem 111
8.3 Group Actions 114
8.4 The Sylow Theorems 116
8.5 Semidirect Products 118
8.6 An Application to Combinatorics 119
9 Series of Subgroups 122
9.1 The Jordan-Holder Theorem 122
9.2 Solvable Groups 124
9.3 Nilpotent Groups 127
10 Galois Theory 130
10.1 Galois Groups and Separability 130
10.2 The Main Theorem of Galois Theory 134
10.3 Insolvability of Polynomials 138
10.4 Cyclotomic Polynomials and Wedderburn's Theorem 140
11 Finiteness Conditions for Rings and Modules 142
11.1 Wedderburn's Theorem 142
11.2 The Wedderburn-Artin Theorem 143
Appendices 147
Appendix A: Complex Numbers 147
Appendix B: Matrix Arithmetic 148
Appendix C: Zorn's Lemma 149
