Solutions Manual to accompany Introduction to Abstract Algebra, 4e
by Nicholson, W. KeithEdition: 4th
Format: Paperback
Pub. Date: 2012-05-15
Publisher(s): Wiley
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Summary
This Fourth Edition of Introduction to Abstract Algebra is a self-contained introduction to the basic structures of abstract algebra: groups, rings, and fields. This book is intended for a one or two semester abstract algebra course. The writing style is appealing to students, and great effort has been made to motivate and be very clear about how the topics and applications relate to one another. Over 500 solved examples are included to aid reader comprehension as well as to demonstrate how results in the theory are obtained. Many applications (particularly to coding theory, cryptography, and to combinatorics) are provided to illustrate how the abstract structures relate to real-world problems. In addition, historical notes and biographies of mathematicians put the subject into perspective. Abstract thinking is difficult when first encountered and this is addressed in this book by presenting concrete examples (induction, number theory, integers modulo n, permutations) before the abstract structures are defined. With this approach, readers can complete computations immediately using concepts that will be seen again later in the abstract setting. Special topics such as symmetric polynomials, nilpotent groups, and finite dimensional algebras are also discussed.
Author Biography
W. KEITH NICHOLSON, PhD, is Professor in the Department of Mathematics and Statistics at the University of Calgary, Canada. He has published extensively in his areas of research interest, which include clean rings, morphic rings and modules, and quasi-morphic rings.
Table of Contents
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
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
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