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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

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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“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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