Introduction to Abstract Algebra, Solutions Manual, 4th EditionISBN: 9781118288153
160 pages
May 2012

Praise for the Third Edition
". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."—Zentralblatt MATH
The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately begin to perform computations using abstract concepts that are developed in greater detail later in the text.
The Fourth Edition features important concepts as well as specialized topics, including:

The treatment of nilpotent groups, including the Frattini and Fitting subgroups

Symmetric polynomials

The proof of the fundamental theorem of algebra using symmetric polynomials

The proof of Wedderburn's theorem on finite division rings

The proof of the WedderburnArtin theorem
Throughout the book, worked examples and realworld problems illustrate concepts and their applications, facilitating a complete understanding for readers regardless of their background in mathematics. A wealth of computational and theoretical exercises, ranging from basic to complex, allows readers to test their comprehension of the material. In addition, detailed historical notes and biographies of mathematicians provide context for and illuminate the discussion of key topics. A solutions manual is also available for readers who would like access to partial solutions to the book's exercises.
Introduction to Abstract Algebra, Fourth Edition is an excellent book for courses on the topic at the upperundergraduate and beginninggraduate levels. The book also serves as a valuable reference and selfstudy tool for practitioners in the fields of engineering, computer science, and applied mathematics.
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 pGroups 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 JordanH¨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 WedderburnArtin Theorem 143
Appendices 147
Appendix A: Complex Numbers 147
Appendix B: Matrix Arithmetic 148
Appendix C: Zorn's Lemma 149