Introduction to Essential Algebraic Structures (eBook)

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