Introduction to Essential Algebraic Structures

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A reader-friendly introduction to modern algebra with important examples from various areas of mathematics Featuring a clear and concise approach, An Introduction to Essential Algebraic Structures presents an integrated approach to basic concepts of modern algebra and highlights topics that play a central role in various branches of mathematics. The authors discuss key topics of abstract and modern algebra including sets, number systems, groups, rings, and fields. The book begins with an exposition of the elements of set theory and moves on to cover the main ideas and branches of abstract algebra. In addition, the book includes: Numerous examples throughout to deepen readers knowledge of the presented material An exercise set after each chapter section in an effort to build a deeper understanding of the subject and improve knowledge retention Hints and answers to select exercises at the end of the book A supplementary website with an Instructors Solutions manual An Introduction to Essential Algebraic Structures is an excellent textbook for introductory courses in abstract algebra as well as an ideal reference for anyone who would like to be more familiar with the basic topics of abstract algebra.

Martyn R. Dixon, PhD, is Professor in the Department of Mathematics at the University of Alabama. Dr. Dixon is the author of over 70 journal articles and two books, including Algebra and Number Theory: An Integrated Approach, also by Wiley.

Leonid A. Kurdachenko, PhD, is Distinguished Professor and Chair of the Department of Algebra at the University of Dnepropetrovsk, Ukraine. Dr. Kurdachenko has authored over 200 journal articles as well as six books, including Algebra and Number Theory: An Integrated Approach, also by Wiley.

Igor Ya. Subbotin, PhD, is Professor in the Department of Mathematics and Natural Sciences at National University in Los Angeles, California. Dr. Subbotin is the author of over 100 journal articles and six books, including Algebra and Number Theory: An Integrated Approach, also by Wiley.


Preface vii

1 Sets 1

1.1 Operations on Sets, 1

Exercise Set 1.1, 7

1.2 Set Mappings, 9

Exercise Set 1.2, 15

1.3 Products of Mappings and Permutations, 16

Exercise Set 1.3, 26

1.4 Operations on Matrices, 28

Exercise Set 1.4, 35

1.5 Binary Algebraic Operations and Equivalence Relations, 37

Exercise Set 1.5, 47

2 Numbers 51

2.1 Some Properties of Integers: Mathematical Induction, 51

Exercise Set 2.1, 55

2.2 Divisibility, 56

Exercise Set 2.2, 63

2.3 Prime Factorization: The Fundamental Theorem of Arithmetic, 64

Exercise Set 2.3, 67

2.4 Rational Numbers, Irrational Numbers, and Real Numbers, 68

Exercise Set 2.4, 76

3 Groups 79

3.1 Groups and Subgroups, 79

Exercise Set 3.1, 93

3.2 Cosets and Normal Subgroups, 94

Exercise Set 3.2, 106

3.3 Factor Groups and Homomorphisms, 108

Exercise Set 3.3, 116

4 Rings 119

4.1 Rings, Subrings, Associative Rings, 119

Exercise Set 4.1, 131

4.2 Rings of Polynomials, 133

Exercise Set 4.2, 142

4.3 Ideals and Quotient Rings, 143

Exercise Set 4.3, 153

4.4 Homomorphisms of Rings, 155

Exercise Set 4.4, 165

5 Fields 169

5.1 Fields: Basic Properties and Examples, 169

Exercise Set 5.1, 180

5.2 Some Field Extensions, 182

Exercise Set 5.2, 187

5.3 Fields of Algebraic Numbers, 187

Exercise Set 5.3, 196

Hints and Answers to Selected Exercises 199

Chapter 1, 199

Chapter 2, 205

Chapter 3, 210

Chapter 4, 214

Chapter 5, 222

Index 225

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Introduction to Essential Algebraic Structures
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