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Kurzbeschreibung This is a self-contained introduction to the basic structures of abstract algebra and its applications. Classroom-tested over several decades, the book is self-contained and is ideal for self-study. The author has thoroughly reviewed and revised the book and has also significantly added to the discussion on modules over principle ideal domains. Hundreds of exercises, with varied levels of difficulty, are included, along with new exercises and additional answers. This book is intended for a one- or two-semester abstract algebra course, and also serves as a self-study guide and reference for students taking Ph.D. qualifying exams.
Aus dem Inhalt 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
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
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-H¨older 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
