Algebra (Prindle, Weber and Schmidt Series in Advanced Mathematics)

1.1 Properties of Functions......Page 9
1.2 Factorizations of Functions......Page 11
1.3 Relations......Page 14
1.4 Equivalence Relations......Page 15
1.5 Generating an Equivalence Relation......Page 18
1.6 Cartesian Products......Page 19
1.7 Formalities about Functions......Page 21
2.1 Groups and Monoids......Page 23
2.2 Subgroups......Page 27
2.3 The Subgroups of the Integers......Page 32
2.4 Finite Cyclic Groups: Modular Arithmetic......Page 35
2.5 Homomorphisms and Isomorphisms......Page 37
2.6 The Classification Problem......Page 43
2.7 The Group of Rotations of the Plane......Page 45
2.8 The Dihedral Groups......Page 46
2.9 Quaternions......Page 48
2.10 Direct Products......Page 50
3 G-sets and Counting......Page 54
3.1 Symmetric Groups: Cayley’s Theorem......Page 55
3.2 Cosets and Index: Lagrange’s Theorem......Page 59
3.3 G-sets and Orbits......Page 63
3.4 Supports of Permutations......Page 72
3.5 Cycle Structure......Page 74
3.6 Conjugation and Other Automorphisms......Page 79
3.7 Conjugating Subgroups: Normality......Page 85
4 Normality and Factor Groups......Page 90
4.1 The Noether Isomorphism Theorems......Page 91
4.2 Simple Groups......Page 98
4.3 The Jordan–Hölder Theorem......Page 104
4.4 Abelian Groups: the Fundamental Theorem......Page 106
4.5 The Automorphisms of a Cyclic Group......Page 113
4.6 Semidirect Products......Page 119
4.7 Extensions......Page 127
5 Sylow Theory, Solvability, and Classification......Page 142
5.1 Cauchy’s Theorem......Page 144
5.2 p-Groups......Page 145
5.3 Sylow Subgroups......Page 149
5.4 Commutator Subgroups and Abelianization......Page 157
5.5 Solvable Groups......Page 158
5.6 Hall’s Theorem......Page 161
5.7 Nilpotent Groups......Page 164
5.8 Matrix Groups......Page 166
6 Categories in Group Theory......Page 171
6.1 Categories......Page 172
6.2 Functors......Page 175
6.3 Universal Mapping Properties: Products and Coproducts......Page 179
6.4 Pushouts and Pullbacks......Page 184
6.5 Infinite Products and Coproducts......Page 191
6.6 Free Functors......Page 194
6.7 Generators and Relations......Page 197
6.8 Direct and Inverse Limits......Page 199
6.9 Natural Transformations and Adjoints......Page 203
6.10 General Limits and Colimits......Page 206
7 Rings and Modules......Page 209
7.1 Rings......Page 210
7.2 Ideals......Page 223
7.3 Polynomials......Page 230
7.4 Symmetry of Polynomials......Page 242
7.5 Group Rings and Monoid Rings......Page 247
7.6 Ideals in Commutative Rings......Page 253
7.7 Modules......Page 258
7.8 Chain Conditions......Page 276
7.9 Vector Spaces......Page 281
7.10 Matrices and Transformations......Page 285
7.11 Rings of Fractions......Page 292
8 P.I.D.s and Field Extensions......Page 303
8.1 Euclidean Rings, P.I.D.s, and U.F.D.s......Page 304
8.2 Algebraic Extensions......Page 316
8.3 Transcendence Degree......Page 322
8.4 Algebraic Closures......Page 325
8.5 Criteria for Irreducibility......Page 328
8.6 The Frobenius......Page 332
8.7 Repeated Roots......Page 333
8.8 Cyclotomic Polynomials......Page 335
8.9 Modules over P.I.D.s......Page 340
9 Radicals, Tensor Products, and Exactness......Page 349
9.1 Radicals......Page 350
9.2 Primary Decomposition......Page 354
9.3 The Nullstellensatz and the Prime Spectrum......Page 359
9.4 Tensor Products......Page 373
9.5 Tensor Products and Exactness......Page 385
9.6 Tensor Products of Algebras......Page 391
9.7 The Hom Functors......Page 394
9.8 Projective Modules......Page 400
9.9 The Grothendieck Construction: K₀......Page 406
9.10 Tensor Algebras and Their Relatives......Page 414
10.1 Traces......Page 424
10.2 Multilinear alternating forms......Page 426
10.3 Properties of determinants......Page 432
10.4 The characteristic polynomial......Page 437
10.5 Eigenvalues and eigenvectors......Page 440
10.6 The classification of matrices......Page 442
10.7 Jordan canonical form......Page 448
10.8 Generators for matrix groups......Page 451
10.9 K₁......Page 455
11 Galois Theory......Page 458
11.1 Embeddings of Fields......Page 459
11.2 Normal Extensions......Page 462
11.3 Finite Fields......Page 466
11.4 Separable Extensions......Page 468
11.5 Galois Theory......Page 473
11.6 The Fundamental Theorem of Algebra......Page 482
11.7 Cyclotomic Extensions......Page 484
11.8 n-th Roots......Page 488
11.9 Cyclic Extensions......Page 493
11.10 Kummer Theory......Page 496
11.11 Solvable Extensions......Page 501
11.12 The General Equation......Page 506
11.13 Normal Bases......Page 508
11.14 Norms and Traces......Page 510
12 Hereditary and Semisimple Rings......Page 514
12.1 Maschke’s Theorem and Projectives......Page 515
12.2 Semisimple Rings......Page 520
12.3 Jacobson Semisimplicity......Page 529
12.4 Homological Dimension......Page 534
12.5 Hereditary Rings......Page 537
12.6 Dedekind Domains......Page 539
12.7 Integral Dependence......Page 547
Bibliography......Page 558