Problems in Abstract Algebra

Cover......Page 1
Title page......Page 2
Contents......Page 4
Preface......Page 8
Introduction......Page 10
0.1. Notation......Page 12
0.2. Zorn’s Lemma......Page 14
Chapter 1. Integers and Integers mod ......Page 16
2.1. Groups, subgroups, and cosets......Page 22
2.2. Group homomorphisms and factor groups......Page 34
2.3. Group actions......Page 41
2.4. Symmetric and alternating groups......Page 45
2.5. -groups......Page 50
2.6. Sylow subgroups......Page 52
2.7. Semidirect products of groups......Page 53
2.8. Free groups and groups by generators and relations......Page 62
2.9. Nilpotent, solvable, and simple groups......Page 67
2.10. Finite abelian groups......Page 75
3.1. Rings, subrings, and ideals......Page 82
3.2. Factor rings and ring homomorphisms......Page 98
3.3. Polynomial rings and evaluation maps......Page 106
3.4. Integral domains, quotient fields......Page 109
3.5. Maximal ideals and prime ideals......Page 112
3.6. Divisibility and principal ideal domains......Page 116
3.7. Unique factorization domains......Page 124
4.1. Vector spaces and linear dependence......Page 134
4.2. Linear transformations and matrices......Page 141
4.3. Dual space......Page 148
4.4. Determinants......Page 151
4.5. Eigenvalues and eigenvectors, triangulation and diagonalization......Page 159
4.6. Minimal polynomials of a linear transformation and primary decomposition......Page 164
4.7. -cyclic subspaces and -annihilators......Page 170
4.8. Projection maps......Page 173
4.9. Cyclic decomposition and rational and Jordan canonical forms......Page 176
4.10. The exponential of a matrix......Page 186
4.11. Symmetric and orthogonal matrices over \R......Page 189
4.12. Group theory problems using linear algebra......Page 196
Chapter 5. Fields and Galois Theory......Page 200
5.1. Algebraic elements and algebraic field extensions......Page 201
5.2. Constructibility by compass and straightedge......Page 208
5.3. Transcendental extensions......Page 211
5.4. Criteria for irreducibility of polynomials......Page 214
5.5. Splitting fields, normal field extensions, and Galois groups......Page 217
5.6. Separability and repeated roots......Page 225
5.7. Finite fields......Page 232
5.8. Galois field extensions......Page 235
5.9. Cyclotomic polynomials and cyclotomic extensions......Page 243
5.10. Radical extensions, norms, and traces......Page 253
5.11. Solvability by radicals......Page 262
Suggestions for Further Reading......Page 266
Bibliography......Page 268
Index of Notation......Page 270
Subject and Terminology Index......Page 276
Back Cover......Page 290