Abstract Algebra

Cover Page......Page 1
Author's note......Page 2
Introduction......Page 3
Contents......Page 5
1.1 Unique factorization......Page 11
1.2 Irrationalities......Page 15
1.3 Z/m, the integers mod m......Page 16
1.4 Fermat’s Little Theorem......Page 18
1.5 Sun-Ze’s theorem......Page 21
1.6 Worked examples......Page 22
2.1 Groups......Page 27
2.2 Subgroups, Lagrange’s theorem......Page 29
2.3 Homomorphisms, kernels, normal subgroups......Page 32
2.4 Cyclic groups......Page 34
2.5 Quotient groups......Page 36
2.6 Groups acting on sets......Page 38
2.7 The Sylow theorem......Page 41
2.8 Trying to classify finite groups, part I......Page 44
2.9 Worked examples......Page 52
3.1 Rings, fields......Page 57
3.2 Ring homomorphisms......Page 60
3.3 Vectorspaces, modules, algebras......Page 62
3.4 Polynomial rings I......Page 64
4.1 Divisibility and ideals......Page 71
4.2 Polynomials in one variable over a field......Page 72
4.3 Ideals and quotients......Page 75
4.4 Ideals and quotient rings......Page 78
4.7 Fermat-Euler on sums of two squares......Page 79
4.8 Worked examples......Page 83
5.1 Some simple results......Page 89
5.2 Bases and dimension......Page 90
5.3 Homomorphisms and dimension......Page 92
6.1 Adjoining things......Page 95
6.2 Fields of fractions, fields of rational functions......Page 98
6.3 Characteristics, finite fields......Page 100
6.4 Algebraic field extensions......Page 102
6.5 Algebraic closures......Page 106
7.1 Irreducibles over a finite field......Page 109
7.2 Worked examples......Page 112
8.1 Multiple factors in polynomials......Page 115
8.2 Cyclotomic polynomials......Page 117
8.3 Examples......Page 120
8.5 Infinitude of primes p = 1 mod n......Page 123
8.6 Worked examples......Page 124
9.1 Uniqueness......Page 129
9.2 Frobenius automorphisms......Page 130
9.3 Counting irreducibles......Page 133
10.1 The structure theorem......Page 135
10.2 Variations......Page 136
10.3 Finitely-generated abelian groups......Page 138
10.4 Jordan canonical form......Page 140
10.5 Conjugacy versus k[x]-module isomorphism......Page 144
10.6 Worked examples......Page 151
11.1 Free modules......Page 161
11.2 Finitely-generated modules over a domain......Page 165
11.3 PIDs are UFDs......Page 168
11.4 Structure theorem, again......Page 169
11.6 Submodules of free modules......Page 171
12.1 Gauss’ lemma......Page 175
12.2 Fields of fractions......Page 177
12.3 Worked examples......Page 179
13.1 Cycles, disjoint cycle decompositions......Page 185
13.3 Worked examples......Page 186
14.1 Sets......Page 191
14.2 Posets, ordinals......Page 193
14.3 Transfinite induction......Page 197
14.5 Comparison of infinities......Page 198
14.6 Example: transfinite Lagrange replacement......Page 200
14.7 Equivalents of the Axiom of Choice......Page 201
15.1 The theorem......Page 203
15.2 First examples......Page 204
15.3 A variant: discriminants......Page 206
16.1 Eisenstein’s irreducibility criterion......Page 209
16.2 Examples......Page 210
17.1 Vandermonde determinants......Page 213
17.2 Worked examples......Page 216
18.1 Cyclotomic polynomials over Z......Page 221
18.2 Worked examples......Page 223
19.1 Another proof of cyclicness......Page 229
19.3 Q with roots of unity adjoined......Page 230
19.4 Solution in radicals, Lagrange resolvents......Page 237
19.5 Quadratic fields, quadratic reciprocity......Page 240
19.6 Worked examples......Page 244
20.1 Prime-power cyclotomic polynomials over Q......Page 253
20.2 Irreducibility of cyclotomic polynomials over Q......Page 255
20.4 Worked examples......Page 256
21.1 Euler’s theorem and the zeta function......Page 271
21.2 Dirichlet’s theorem......Page 273
21.3 Dual groups of abelian groups......Page 276
21.4 Non-vanishing on Re(s) = 1......Page 278
21.5 Analytic continuations......Page 279
21.6 Dirichlet series with positive coefficients......Page 280
22 Galois theory......Page 283
22.1 Field extensions, imbeddings, automorphisms......Page 284
22.2 Separable field extensions......Page 285
22.3 Primitive elements......Page 287
22.4 Normal field extensions......Page 288
22.5 The main theorem......Page 290
22.7 Basic examples......Page 292
22.8 Worked examples......Page 293
23.1 Galois’ criterion......Page 304
23.3 Solving cubics by radicals......Page 306
23.4 Worked examples......Page 309
24.1 Eigenvectors, eigenvaluesv......Page 314
24.2 Diagonalizability, semi-simplicity......Page 317
24.3 Commuting endomorphisms ST = TS24.3 Commuting endomorphisms ST = TS......Page 319
24.4 Inner product spaces......Page 320
24.6 Unitary operators......Page 325
24.7 Spectral theorems......Page 326
24.8 Corollaries of the spectral theorem......Page 327
24.9 Worked examples......Page 329
25.1 Dual vectorspaces......Page 336
25.2 First example of naturality......Page 340
25.3 Bilinear forms......Page 341
25.4 Worked examples......Page 344
26.1 Prehistory......Page 352
26.2 Definitions......Page 354
26.3 Uniqueness and other properties......Page 355
26.4 Existence......Page 359
27.1 Desiderata......Page 362
27.2 Definitions, uniqueness, existence......Page 363
27.3 First examples......Page 367
27.4 Tensor products f ⊗ g of maps......Page 370
27.5 Extension of scalars, functoriality, naturality......Page 371
27.6 Worked examples......Page 374
28.1 Desiderata......Page 386
28.2 Definitions, uniqueness, existence......Page 387
28.3 Some elementary facts......Page 390
28.4 Exterior powers Λn(f) of maps......Page 391
28.5 Exterior powers of free modules......Page 392
28.6 Determinants revisited......Page 395
28.7 Minors of matrices......Page 396
28.8 Uniqueness in the structure theorem......Page 397
28.9 Cartan’s lemma......Page 398
28.10 Cayley-Hamilton theorem......Page 399
28.11 Worked examples......Page 401