Introduction to Modern Algebra

Description

Contents Preface vii 1 . Basic Structures 1 1.1 Sets and Functions 1 Exercises 14 Glossary 15 1.2 Algebraic Structures 16 Exercises 25 Glossary 30 1.3 Permutation Groups 31 Exercises 52 Glossary 54 2. Groups 56 2.1 Isomorphism Theorems 56 Exercises 33 Glossary 86 2.2 Group Actions and the Sylow Theorems 86 Exercises 106 Glossary 109 2.3 Some Additional Topics 109 Exercises 123 Glossary 123 3. Rings and Fields 124 3.1 Basic Facts 124 Exercises 136 Glossary 138 3.2 Introduction to Polynomial Rings 139 Exercises 170 Glossary 172 3.3 Unique Factorization Domains 1'13 Exercises 190 Glossary 193 3.4 Polynomial Factorization 194 Exercises 216 Glossary 224 3.5 Polynomials and Resultants 225 Exercises 255 Glossary 257 3.6 Applications to Geometric Constructions 257 Exercises 263 Glossary 264 3.7 Galois Theory 264 Exercises 282 Glossary 235 4. Modules and Linear Operators 28? 4.1 The Hermite Normal Form 287 Exercises 316 Glossary 319 4.2 The Smith Normal Form 320 Exercises 344 Glossary 350 4.3 The Structure of Linear Transformations 351 Exercises 381 Glossary 396 4.4 Introduction to Multilinear Algebra 396 Exercises 411 Glossary 412 5. Representations of Groups 413 5.1 Representations 413 Exercises 435 Glossary 439 5.2 Matrix Representations 440 Exercises 474 Glossary 477 Index 479