Solutions Manual To Accompany Introduction To Abstract Algebra, Fourth Edition

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 Wedderburn-Artin theorem

Throughout the book, worked examples and real-world 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 upper-undergraduate and beginning-graduate levels. The book also serves as a valuable reference and self-study tool for practitioners in the fields of engineering, computer science, and applied mathematics.

Peter Ayres was taught by, or worked with, several of Tansley's closest friends. He was for seven years Executive Editor of the New Phytologist, the journal founded by Tansley.? After a career teaching plant physiology and pathology at Lancaster University, his interest in the history of plant sciences has led him to write Harry Marshall Ward and the Fungal Thread of Death and The Aliveness of Plants: The Darwins at the Dawn of Plant Science.

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