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Introduction to Abstract Algebra, Solutions Manual, 4th Edition

ISBN: 978-1-118-28815-3
160 pages
May 2012
Introduction to Abstract Algebra, Solutions Manual, 4th Edition (1118288157) cover image

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.

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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-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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“This could also be an excellent adjunct to more theoretically oriented textbooks used in more intensive courses.”  (Computing Reviews, 5 November 2012)

 

 

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