CONTENTS
Section 0 A very few Preliminaries
0.1 The role of definitions = 1
0.2 Sets = 2
0.3 Partitions and equivalence relations = 3
PART Ⅰ 1 GROUPS
Section 1 Binary Operations
1.1 Motivation = 9
1.2 Definition and properties = 9
1.3 Tables = 11
1.4 Some words of warning = 12
Section 2 Groups
2.1 Motivation = 17
2.2 Definition and elementary properties = 18
2.3 Finite groups and group tables = 21
Section 3 Subgroups
3.1 Notation and terminology = 27
3.2 Subsets and subgroups = 28
3.3 Cyclic subgroups = 30
Section 4 Permutations Ⅰ
4.1 Functions and permutations = 35
4.2 Groups of permutations = 38
4.3 Two important examples = 40
Section 5 Permutations Ⅱ
5.1 Cycles and cyclic notation = 45
5.2 Even and odd permutations = 47
5.3 The alternating groups = 49
Section 6 Cyclic Groups
6.1 Elementary properties = 52
6.2 The classification of cyclic groups = 54
6.3 Subgroups of finite cyclic groups = 56
Section 7 Isomorphism
7.1 Definition and elementary properties = 59
7.2 How to show that groups are isonorphic = 60
7.3 How to show that groups are not isomorphic = 62
7.4 Cayley's theorem = 63
Section 8 Direct Products
8.1 External direct products = 69
8.2 Internal direct products = 73
Section 9 Finitely Generated Abelian Groups
9.1 Generators and torsion = 77
9.2 The fundamental theorem = 79
9.3 Applications = 81
Section 10 Groups in Geometry and Analysis
10.1 Groups in geometry = 84
10.2 Groups in analysis = 88
Section 11 Groups of Cosets
11.1 Introduction = 92
11.2 Cosets = 93
11.3 Applications = 97
Section 12 Normal Subgroups and Factor Groups
12.1 Criteria for the existence of a coset group = 101
12.2 Inner automorphisms and normal subgroups = 102
12.3 Factor groups = 104
12.4 Simple groups = 107
12.5 Applications = 107
Section 13 Homomorphisms
13.1 Definition and elementary properties = 112
13.2 The fundamental homomorphism theorem = 114
13.3 Applications = 116
13.4 The first isomorphism theorem = 117
Section 14 Series of Groups
14.1 Subnormal and normal scries = 121
14.2 The Jordan-H$$\ddot o$$lder theorem = 122
14.3 The center and the .ascending central scries = 125
Section 15 The Sylow Theorems
15.1 Conjugate classes and the class equation = 128
15.2 The Sylow theorems = 129
15.3 Applications to p-groups = 130
15.4 Some proofs = 131
Section 16 Applications of the Sylow Theory
16.1 More applications to p-groups = 135
16.2 Further applications = 136
Section 17 Free Groups
17.1 Words and reduced words = 140
17.2 Free groups = 141
17.3 Homomorphisms of free groups = 142
17.4 Free abelian groups = 143
Section 18 Group Presentations
18.1 Definition = 149
18.2 Isomorphic presentations = 150
18.3 Applications = 151
Section 19 Simplicial Complexes and Homology Groups
19.1 Motivation = 157
19.2 Preliminary notions = 158
19.3 Chains, cycles, and boundaries = 160
19.4 $$%^2$$ ＝ 0 and homology groups = 163
Section 20 Computations of Homology Groups
20.1 Triangulations = 166
20.2 Invariance properties = 166
20.3 Connected and contractible spaces = 167
20.4 Further computations = 169
Section 21 More Homology Computations and Applications
21.1 One-sided surfaces = 174
21.2 The Euler characteristic = 176
21.3 Mappings of spaces = 178
Section 22 Homological Algebra
22.1 Chain complexes and mappings = 183
22.2 Relative homology = 185
22.3 The exact homology sequence of a pair = 188
PART Ⅱ I RINGS AND FIELDS
Section 23 Rings
23.1 Definition and basic properties = 195
23.2 Multiplicative questions ; fields = 197
Section 24 Integral Domains
24.1 Divisors of zero and cancellation = 202
24.2 Integral domains = 204
24.3 The characteristic of a ring = 205
24.4 Fermat's theorem = 205
24.5 Euler's generalization = 206
Section 25 Some Noncommutative Examples
25.1 Matrices over a field = 209
25.2 Rings of endomorphisms = 211
25.3 Group rings and group algebras = 213
25.4 The quaternions = 215
Section 26 The Field of Quotients of an Integral Domain
26.1 The construction = 219
26.2 Uniqueness = 223
Section 27 Our Basic Goal = 227
Section 28 Quotient Rings and Ideals
28.1 Introduction = 230
28.2 Criteria for the existence of a coset ring = 231
28.3 Ideals and quotient rings = 232
Section 29 Homomorphisms of Rings
29.1 Definition and elementary properties = 236
29.2 Maximal and prime ideals = 238
29.3 Prime fields = 240
Section 30 Rings of Polynomials
30.1 Polynomials in an indeterminate = 244
30.2 The evaluation homomorphisms = 247
30.3 The new approach = 250
Section 31 Factorization of Polynomials Over a Field
31.1 The division algorithm in F[x] = 254
31.2 Irreducible polynomials = 258
31.3 Ideal structure in F[x] = 261
31.4 Uniqueness of factorization in F[x] = 262
Section 32 Unique Factorization Domains
32.1 introduction = 266
32.2 Every PID is UFD = 271
32.3 If D is a UFD, then D[x] is a UFD = 271
Section 33 Euclidean Domains
33.1 Introduction and definition = 278
33.2 Arithmetic in Euclidean domains = 279
Section 34 Gaussian Integers and Norms
34.1 Gaussian integers = 286
34.2 Muliplicative norms = 288
Section 35 Introduction to Extension Fields
35.1 Our basic goal achieved = 293
35.2 Algebraic and transcendental elements = 295
35.3 The irreducible polynomial for a over F = 296
35.4 Simple extensions = 297
Section 36 Vector Spaces
36.1 Definition and elementary properties = 302
36.2 Linear independence and bases = 304
36.3 Dimension = 305
36.4 An application to field theory = 307
Section 37 Further Algebraic Structures
37.1 Groups with operators = 311
37.2 Modules = 313
37.3 Algebras = 314
Section 38 Algebraic Extensions
38.1 Finite extensions = 317
38.2 Algebraically closed fields and algebraic closures = 321
38.3 The existence of an algebraic closure = 322
Section 39 Geometric Constructions
39.1 Constructible numbers = 328
39.2 The impossibility of certain constructions = 331
Section 40 Automorphisms of Fields
40.1 The basic isomorphisms of algebraic field theory = 335
40.2 Automorphisms and fixed fields = 338
40.3 The Frobenius automorphism = 341
Section 41 The Isomorphism Extension Theorem
41.1 The extension theorem = 345
41.2 The index of a field extension = 347
41.3 Proof of the extension theorem = 349
Section 42 Splitting Fields = 353
Section 43 Separable Extensions
43.1 Multiplicity of zeros of a polynomial = 358
43.2 Separable extensions = 360
43.3 Perfect fields = 361
43.4 The primitive element theorem = 363
Section 44 Totally Inseparable Extensions
44.1 Totally inseparable extensions = 366
44.2 Separable closures = 367
Section 45 Finite Fields
45.1 The structure of a finite field = 370
45.2 The existence of GF ($$p^n$$) = 372
Section 46 Galois Theory
46.1 R$$\acute e$$sum$$\acute e$$ = 375
46.2 Normal extensions = 375
46.3 The main theorem = 377
46.4 Galois gro %s over finite fields = 379
46.5 Proof of the %ain theorem completed = 380
Section 47 Illustrations of Galois Theory
47 1 Symmetric functions = 385
47 2 Examples = 387
Section 48 Cyclotomic Extensions
48.1 The Galois group of a cyclotomic extension = 393
48.2 Constructible polygons = 395
Section 49 Insolvability of the Quintic
49.1 The problem = 400
49.2 Extensions by radicals = 400
49.3 The insolvability of the quintic = 402
Bibliography = 407
Answers and Comments = 411
Notations = 443
Index = 447
