Algebra (Pure and Applied Mathematics (Academic Pr))

This graduate-level text is intended for initial courses in algebra that proceed at a faster pace than undergraduate-level courses. Exercises appear throughout the text, clarifying concepts as they arise; additional exercises, varying widely in difficulty, are included at the ends of the chapters. Subjects include groups, rings, fields and Galois theory. 1983 edition. Includes 11 figures. Appendix. References. Index. Cover......Page 1 Series......Page 2 Title page......Page 3 Date-line......Page 4 Contents......Page 5 Preface......Page 7 List of Symbols......Page 9 Introduction......Page 13 1. Groups, Subgroups, and Homomorphisms......Page 15 2. Permutation Groups......Page 26 3. The Symmetric and Alternating Groups......Page 30 4. The Sylow Theorems......Page 33 5. Solvable Groups, Normal and Subnormal Series......Page 36 6. Products......Page 41 7. Nilpotent Groups......Page 43 8. Finite Abelian Groups......Page 46 9. Free Groups......Page 47 10. Generators and Relations......Page 51 11. Some Finite Groups Classified......Page 54 12. Further Exercises......Page 55 1. Preliminaries: Ideals and Homomorphisms......Page 61 2. The Field of Fractions of an Integral Domain......Page 66 3. Polynomials......Page 68 4. Polynomials in Several lndeterminates......Page 72 5. Divisibility and Factorization......Page 75 6. The Chinese Remainder Theorem......Page 85 7. The Hilbert Basis Theorem......Page 87 8. Further Exercises......Page 89 1. Field Extensions......Page 93 2. The Fundamental Theorem of Galois Theory......Page 100 3. Normality and Separability......Page 104 4. The Galois Theory of Equations......Page 110 5. Symmetric Functions......Page 116 6. Geometrical Constructions......Page 123 7. Norm and Trace......Page 129 8. Further Exercises......Page 133 1. Preliminaries......Page 139 2. Direct Sums, Free Modules......Page 143 3. Finitely Generated Modules over a PID......Page 148 4. Applications to Linear Algebra......Page 154 5. Computations......Page 159 6. Tensor Products......Page 172 7. Further Exercises......Page 178 1. Preliminaries......Page 186 2. The Jacobson Radical......Page 190 3. The Density Theorem......Page 194 4. Artinian Rings......Page 197 5. Further Exercises......Page 203 1. Infinite Abelian Groups......Page 208 2. Polya-Redfield Enumeration......Page 214 3. PSL($V$)......Page 224 4. Integral Dependence and Dedekind Domains......Page 231 5. Transcendental Field Extensions......Page 244 6. Valuations and $p$-adic Numbers......Page 252 7. Real Fields and Sturm's Theorem......Page 266 8. Representations and Characters of Finite Groups......Page 274 9. Some Galois Groups......Page 289 Appendix Zorn's Lemma......Page 303 References......Page 305 Index......Page 307