Algebra and Number Theory: An Integrated Approach

Preface ix

Chapter 1 Sets 1

1.1 Operations on Sets 1

Exercise Set 1.1 6

1.2 Set Mappings 8

Exercise Set 1.2 19

1.3 Products of Mappings 20

Exercise Set 1.3 26

1.4 Some Properties of Integers 28

Exercise Set 1.4 39

Chapter 2 Matrices and Determinants 41

2.1 Operations on Matrices 41

Exercise Set 2.1 52

2.2 Permutations of Finite Sets 54

Exercise Set 2.2 64

2.3 Determinants of Matrices 66

Exercise Set 2.3 77

2.4 Computing Determinants 79

Exercise Set 2.4 91

2.5 Properties of the Product of Matrices 93

Exercise Set 2.5 103

Chapter 3 Fields 105

3.1 Binary Algebraic Operations 105

Exercise Set 3.1 118

3.2 Basic Properties of Fields 119

Exercise Set 3.2 129

3.3 The Field of Complex Numbers 130

Exercise Set 3.3 144

Chapter 4 Vector Spaces 145

4.1 Vector Spaces 146

Exercise Set 4.1 158

4.2 Dimension 159

Exercise Set 4.2 172

4.3 The Rank of a Matrix 174

Exercise Set 4.3 181

4.4 Quotient Spaces 182

Exercise Set 4.4 186

Chapter 5 Linear Mappings 187

5.1 Linear Mappings 187

Exercise Set 5.1 199

5.2 Matrices of Linear Mappings 200

Exercise Set 5.2 207

5.3 Systems of Linear Equations 209

Exercise Set 5.3 215

5.4 Eigenvectors and Eigenvalues 217

Exercise Set 5.4 223

Chapter 6 Bilinear Forms 226

6.1 Bilinear Forms 226

Exercise Set 6.1 234

6.2 Classical Forms 235

Exercise Set 6.2 247

6.3 Symmetric Forms over R 250

Exercise Set 6.3 257

6.4 Euclidean Spaces 259

Exercise Set 6.4 269

Chapter 7 Rings 272

7.1 Rings, Subrings, and Examples 272

Exercise Set 7.1 287

7.2 Equivalence Relations 288

Exercise Set 7.2 295

7.3 Ideals and Quotient Rings 297

Exercise Set 7.3 303

7.4 Homomorphisms of Rings 303

Exercise Set 7.4 313

7.5 Rings of Polynomials and Formal Power Series 315

Exercise Set 7.5 327

7.6 Rings of Multivariable Polynomials 328

Exercise Set 7.6 336

Chapter 8 Groups 338

8.1 Groups and Subgroups 338

Exercise Set 8.1 348

8.2 Examples of Groups and Subgroups 349

Exercise Set 8.2 358

8.3 Cosets 359

Exercise Set 8.3 364

8.4 Normal Subgroups and Factor Groups 365

Exercise Set 8.4 374

8.5 Homomorphisms of Groups 375

Exercise Set 8.5 382

Chapter 9 Arithmetic Properties of Rings 384

9.1 Extending Arithmetic to Commutative Rings 384

Exercise Set 9.1 399

9.2 Euclidean Rings 400

Exercise Set 9.2 404

9.3 Irreducible Polynomials 406

Exercise Set 9.3 415

9.4 Arithmetic Functions 416

Exercise Set 9.4 429

9.5 Congruences 430

Exercise Set 9.5 446

Chapter 10 The Real Number System 448

10.1 The Natural Numbers 448

10.2 The Integers 458

10.3 The Rationals 468

10.4 The Real Numbers 477

Answers to Selected Exercises 489

Index 513