Contents of Algebra*
Mark Steinberger

Chapter 1. A Little Set Theory ... 1

1.1. Properties of Functions ... 1
1.2. Factorizations of Functions ... 3
1.3. Relations ... 6
1.4. Equivalence Relations ... 8
1.5. Generating an Equivalence Relation ... 11
1.6. Cartesian Products ... 12
1.7. Formalities about Functions ... 14

Chapter 2. Groups: Basic Definitions and Examples ... 16

2.1. Groups and Monoids ... 16
2.2. Subgroups ... 20
2.3. The Subgroups of the Integers ... 25
2.4. Finite Cyclic Groups: Modular Arithmetic ... 28
2.5. Homomorphisms and Isomorphisms ... 30
2.6. The Classification Problem ... 36
2.7. The Group of Rotations of the Plane ... 38
2.8. The Dihedral Groups ... 40
2.9. Quaternions ... 42
2.10. Direct Products ... 44

Chapter 3. G-sets and Counting ... 48

3.1. Symmetric Groups: Cayley's Theorem ... 49
3.2. Cosets and Index: Lagrange's Theorem ... 53
3.3. G-sets and Orbits ... 57
3.4. Supports of Permutations ... 66
3.5. Cycle Structure ... 68
3.6. Conjugation and Other Automorphisms ... 73
3.7. Conjugating Subgroups: Normality ... 79

Chapter 4. Normality and Factor Groups ... 84

4.1. The Noether Isomorphism Theorems ... 85
4.2. Simple Groups ... 92
4.3. The Jordan-Hoelder Theorem ... 98
4.4. Abelian Groups: the Fundamental Theorem ... 101
4.5. The Automorphisms of a Cyclic Group ... 108
4.6. Semidirect Products ... 114
4.7. Extensions ... 123

Chapter 5. Sylow Theory, Solvability, and Classification ... 139

5.1. Cauchy's Theorem ... 141
5.2. p-Groups ... 142
5.3. Sylow Subgroups ... 146
5.4. Commutator Subgroups and Abelianization ... 154
5.5. Solvable Groups ... 156
5.6. Hall's Theorem ... 159
5.7. Nilpotent Groups ... 162
5.8. Matrix Groups ... 165

Chapter 6. Categories in Group Theory ... 169

6.1. Categories ... 170
6.2. Functors ... 173
6.3. Universal Mapping Properties: Products and Coproducts ... 177
6.4. Pushouts and Pullbacks ... 183
6.5. Infinite Products and Coproducts ... 190
6.6. Free Functors ... 193
6.7. Generators and Relations ... 197
6.8. Direct and Inverse Limits ... 199
6.9. Natural Transformations and Adjoints ... 203
6.10. General Limits and Colimits ... 206

Chapter 7. Rings and Modules ... 209

7.1. Rings ... 210
7.2. Ideals ... 223
7.3. Polynomials ... 230
7.4. Symmetry of Polynomials ... 242
7.5. Group Rings and Monoid Rings ... 247
7.6. Ideals in Commutative Rings ... 253
7.7. Modules ... 258
7.8. Chain Conditions ... 277
7.9. Vector Spaces ... 282
7.10. Matrices and Transformations ... 286
7.11. Rings of Fractions ... 294

Chapter 8. P.I.D.s and Field Extensions ... 304

8.1. Euclidean Rings, P.I.D.s, and U.F.D.s ... 305
8.2. Algebraic Extensions ... 317
8.3. Transcendence Degree ... 323
8.4. Algebraic Closures ... 326
8.5. Criteria for Irreducibility ... 329
8.6. The Nullstellensatz and the Prime Spectrum ... 334
8.7. The Frobenius ... 337
8.8. Repeated Roots ... 339
8.9. Cyclotomic Polynomials ... 342
8.10. Modules over P.I.D.s ... 346

Chapter 9. Radicals, Tensor Products, and Exactness ... 356

9.1. Radicals ... 357
9.2. Tensor Products ... 360
9.3. Tensor Products and Exactness ... 372
9.4. Tensor Products of Algebras ... 380
9.5. The Hom Functors ... 383
9.6. Projective Modules ... 388
9.7. The Grothendieck Construction: K_0
9.8. Tensor Algebras and Their Relatives ... 403

Chapter 10. Linear Algebra ... 413

10.1. Traces ... 414
10.2. Multilinear Alternating Forms ... 415
10.3. Properties of Determinants ... 421
10.4. The Characteristic Polynomial ... 427
10.5. Eigenvalues and Eigenvectors ... 430
10.6. The Classification of Matrices ... 432
10.7. Jordan Canonical Form ... 439
10.8. Generators for Matrix Groups ... 442
10.9. K_1 ... 445

Chapter 11. Galois Theory ... 448

11.1. Embeddings of Fields ... 449
11.2. Normal Extensions ... 452
11.3. Finite Fields ... 456
11.4. Separable Extensions ... 458
11.5. Galois Theory ... 463
11.6. The Fundamental Theorem of Algebra ... 473
11.7. Cyclotomic Extensions ... 475
11.8. n-th Roots ... 479
11.9. Cyclic Extensions ... 484
11.10. Kummer Theory ... 488
11.11. Solvable Extensions ... 492
11.12. The General Equation ... 498
11.13. Normal Bases ... 500
11.14. Norms and Traces ... 503

Chapter 12. Hereditary and Semisimple Rings ... 506

12.1. Maschke's Theorem and Projectives ... 507
12.2. Semisimple Rings ... 512
12.3. Jacobson Semisimplicity ... 521
12.4. Homological Dimension ... 526
12.5. Hereditary Rings ... 529
12.6. Dedekind Domains ... 531
12.7. Integral Dependence ... 539

Bibliography ... 552

Index ... 553

*PWS Publishing Co.,1994; Currently available from Brooks/Cole
Mark Steinberger's home page