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CHAPTER1 RUDIMENTS 1

1.1 Sets 1

Classical Problems:Sets 7

Supplemental Exercises:Sets 9

1.2 Mappings 9

INTRODUCTION 11

Classical Problems:Mappings 15

Supplemental Exercises:Mappings 18

1.3 Relations and Operations 19

Classical Problems:Relations and Operations 24

Supplemental Exercises:Relations and Operations 28

1.4 Number Systems 29

1.4.1 The Natural Numbers 29

1.4.2 The Integers 31

1.4.3 The Rational Numbers 36

1.4.4 The Reals 37

1.4.5 The Complex Numbers 38

Classical Problems:Number Systems 39

Supplemental Exercises:Number Systems 49

CHAPTER2 GROUPS 51

2.1 Introduction to Groups 51

Classical Problems:Groups and Subgroups 57

2.2 Working With Groups 63

Classical Problems:Working With Groups 69

2.3 More on Group Structure 79

Classical Problems:More on Group Structure 81

2.4 Supplemental Exercises:Groups 90

CHAPTER 3 PINGS 93

3.1 Basic Ring Structure 93

Classical Problems:Basic Ring Structure 96

3.2 Ring Substructures 102

Classical Problems:Ring Substructures 104

3.3 Specialized Rings 110

Classical Problems:Specialized Rings 113

3.4 Working With Rings 120

Classical Problems:Working With Rings 122

3.5 Notes on Rings 128

3.6 Supplemental Exercises:Rings 129

CHAPTER 4 R-MODULES 131

4.1 Introduction to R-Modules 131

4.2 Notes on Modules 135

4.3 Classical Problems: R-Modules 140

4.4 Supplemental Exercises:R-Modules 144

CHAPTER 5 VECTOR SPACES 145

5.1 Introduction to Vector Spaces 145

5.2 Nots on Vector Spaces 151

5.3 Classical Problems:Vector Spaces 152

5.4 Supplemental Exercises:Vector Spaces 158

CHAPTER 6 INTRODUCTION TO MATRICES 159

6.1 Basic Linear Algebra 159

6.1.1 Basic Structures 159

6.1.2 Notes:Basic Linear Algebra 167

Classical Problems:Matrices 169

6.2.1 Introduction 176

6.2 Matrices in Solving Systems of Equations 176

6.2.2 Examples 180

Classical Problems:Applying Matrices in Solving Systems of Equations 181

6.3 Supplemental Exercises:Matrices 186

CHAPTER 7 POLYNOMIALS 188

7.1 Definitions 188

7.2 Background and Notes:Polynomials 192

7.3 Classical Problems:Polynomials 193

7.4 Supplemental Exercises:Polynomials 196

8.1 Definitions 198

CHAPTER 8 INTRODUCTION TO GALOIS THEORY 198

8.2 Theorems 202

8.3 Background and Notes:Galois Theory 203

8.4 Classical Problems:Extension Fields 206

8.5 Supplemental Exercises:Galois Theory 209

GLOSSARY 215

BIOGRAPHICAL SKETCHES 217

BIBLIOGRAPHY 221

INDEX 223