Algebra

Contents......Page 4
Preface......Page 10
1.1 The Basic Operations......Page 16
1.2 Row Reduction......Page 25
1.3 The Matrix Transpose......Page 32
1.4 Determinants......Page 33
1.5 Permutations......Page 39
1.6 Other Formulas for the Determinant......Page 42
Exercises......Page 46
2.1 Laws of Composition......Page 52
2.2 Groups and Subgroups......Page 55
2.3 Subgroups of the Additive Group of Integers......Page 58
2.4 Cyclic Groups......Page 61
2.5 Homomorphisms......Page 62
2.6 Isomorphisms......Page 66
2.7 Equivalence Relations and Partitions......Page 67
2.8 Cosets......Page 71
2.9 Modular Arithmetic......Page 75
2.10 The Correspondence Theorem......Page 76
2.11 Product Groups......Page 79
2.12 Quotient Groups......Page 81
Exercises......Page 84
3.1 Subspaces of Rn......Page 93
3.2 Fields......Page 95
3.3 Vector Spaces......Page 99
3.4 Bases and Dimension......Page 101
3.5 Computing with Bases......Page 106
3.6 Direct Sums......Page 110
3.7 Infinite-Dimensional Spaces......Page 111
Exercises......Page 113
4.1 The Dimension Formula......Page 117
4.2 The Matrix of a Linear Transformation......Page 119
4.3 Linear Operators......Page 123
4.4 Eigenvectors......Page 125
4.5 The Characteristic Polynomial......Page 128
4.6 Triangular and Diagonal Forms......Page 131
4.7 Jordan Form......Page 135
Exercises......Page 140
5.1 Orthogonal Matrices and Rotations......Page 147
5.2 Using Continuity......Page 153
5.3 Systems of Differential Equations......Page 156
5.4 The Matrix Exponential......Page 160
Exercises......Page 165
6.1 Symmetry of Plane Figures......Page 169
6.2 Isometries......Page 171
6.3 Isometries of the Plane......Page 174
6.4 Finite Groups of Orthogonal Operators on the Plane......Page 178
6.5 Discrete Groups of Isometries......Page 182
6.6 Plane Crystallographic Groups......Page 187
6.7 Abstract Symmetry: Group Operations......Page 191
6.8 The Operation on Cosets......Page 193
6.9 The Counting Formula......Page 195
6.11 Permutation Representations......Page 196
6.12 Finite Subgroups of the Rotation Group......Page 198
Exercises......Page 203
7.2 The Class Equation......Page 210
7.3 p-Groups......Page 212
7.4 The Class Equation of the Icosahedral Group......Page 213
7.5 Conjugation in the Symmetric Group......Page 215
7.7 The Sylow Theorems......Page 218
7.8 Groups of Order 12......Page 223
7.9 The Free Group......Page 225
7.10 Generators and Relations......Page 227
7.11 The Todd-Coxeter Algorithm......Page 231
Exercises......Page 236
8.1 Bilinear Forms......Page 244
8.2 Symmetric Forms......Page 246
8.3 Hermitian Forms......Page 247
8.4 Orthogonality......Page 250
8.5 Euclidean Spaces and Hermitian Spaces......Page 256
8.6 The Spectral Theorem......Page 257
8.7 Conics and Quadrics......Page 260
8.8 Skew-Symmetric Forms......Page 264
8.9 Summary......Page 267
Exercises......Page 269
9.1 The Classical Groups......Page 276
9.2 Interlude: Spheres......Page 278
9.3 The Special Unitary Group SU2......Page 281
9.4 The Rotation Group SO3......Page 284
9.5 One-Parameter Groups......Page 287
9.6 The Lie Algebra......Page 290
9.7 Translation in a Group......Page 292
9.8 Normal Subgroups of SL2......Page 295
Exercises......Page 298
10.1 Definitions......Page 305
10.2 Irreducible Representations......Page 309
10.3 Unitary Representations......Page 311
10.4 Characters......Page 313
10.5 One-Dimensional Characters......Page 318
10.6 The Regular Representation......Page 319
10.7 Schur\'s Lemma......Page 322
10.8 Proof of the Orthogonality Relations......Page 324
10.9 Representations of SU2......Page 326
Exercises......Page 329
11.1 Definition of a Ring......Page 338
11.2 Polynomial Rings......Page 340
11.3 Homomorphisms and Ideals......Page 343
11.4 Quotient Rings......Page 349
11.5 Adjoining Elements......Page 353
11.6 Product Rings......Page 356
11.7 Fractions......Page 357
11.8 Maximal Ideals......Page 359
11.9 Algebraic Geometry......Page 362
Exercises......Page 369
12.1 Factoring Integers......Page 374
11.2 Unique Factorization Domains......Page 375
12.3 Gauss\'s Lemma......Page 382
12.4 Factoring Integer Polynomials......Page 386
12.5 Gauss Primes......Page 391
Exercises......Page 393
13.1 Algebraic Integers......Page 398
13.2 Factoring Algebraic Integers......Page 400
13.3 Ideals in Z[√-5]......Page 402
13.4 Ideal Multiplication......Page 404
13.5 Factoring Ideals......Page 407
13.6 Prime Ideals and Prime Integers......Page 409
13.7 Ideal Classes......Page 411
13.8 Computing the Class Group......Page 414
13.9 Real Quadratic Fields......Page 417
13.10 About Lattices......Page 420
Exercises......Page 423
14.1 Modules......Page 427
14.2 Free Modules......Page 429
14.3 Identities......Page 432
14.4 Diagonalizing Integer Matrices......Page 433
14.5 Generators and Relations......Page 438
14.6 Noetherian Rings......Page 441
14.7 Structure of Abelian Groups......Page 444
14.8 Application to Linear Operators......Page 447
14.9 Polynomial Rings in Several Variables......Page 451
Exercises......Page 452
15.1 Examples of Fields......Page 457
15.2 Algebraic and Transcendental Fields......Page 458
15.3 The Degree of a Field Extension......Page 461
15.4 Finding the Irreducible Polynomial......Page 464
15.5 Ruler and Compass Constructions......Page 465
15.6 Adjoining Roots......Page 471
15.7 Finite Fields......Page 474
15.8 Primitive Elements......Page 477
15.9 Function Fields......Page 478
15.10 The Fundamental Theorem of Algebra......Page 486
Exercises......Page 487
16.1 Symmetric Functions......Page 492
16.2 The Discriminant......Page 496
16.3 Splitting Fields......Page 498
16.4 Isomorphisms of Field Extensions......Page 499
16.5 Fixed Fields......Page 501
16.6 Galois Extensions......Page 503
16.7 The Main Theorem......Page 504
16.8 Cubic Equations......Page 507
16.9 Quartic Equations......Page 508
16.10 Roots of Unity......Page 512
16.11 Kummer Extensions......Page 515
16.12 Quintic Equations......Page 517
Exercises......Page 520
A.1 About Proofs......Page 528
A.2 The Integers......Page 531
A.3 Zorn\'s Lemma......Page 533
A.4 The Implicit Function Theorem......Page 534
Bibliography......Page 538
Notation......Page 540
Index......Page 544