0 Sets and Relations 1
Ⅰ GROUPS AND SUBGROUPS 11
1 Introduction and Examples 11
2 Binary Operations 20
3 Isomorphic Binary Structures 28
4 Groups 36
5 Subgroups 49
6 Cyclic Groups 59
7 Generating Sets and Cayley Digraphs 68
Ⅱ PERMUTATIONS,COSETS,AND DIRECT PRODUCTS 75
8 Groups of Permutations 75
9 Orbits,Cycles,and the Alternating Groups 87
10 Cosets and the Theorem of Lagrange 96
11 Direct Products and Finitely Generated Abelian Groups 104
12 Plane Isometries 114
Ⅲ HOMOMORPHISMS AND FACTOR GROUPS 125
13 Homomorphisms 125
14 Factor Groups 135
15 Factor-Group Computations and Simple Groups 144
16 Group Action on a Set 154
17 Applications of G-Sets to Counting 161
Ⅳ RINGS AND FIELDS 167
18 Rings and Fields 167
19 Integral Domains 177
20 Fermat's and Euler's Theorems 184
21 The Field of Quotients of an Integral Domain 190
22 Rings of Polynomials 198
23 Factorization of Polynomials over a Field 209
24 Noncommutative Examples 220
25 Ordered Rings and Fields 227
Ⅴ IDEALS AND FACTOR RINGS 237
26 Homomorphisms and Factor Rings 237
27 Prime and Maximal Ideals 245
28 Gr？bner Bases for Ideals 254
Ⅵ EXTENSION FIELDS 265
29 Introduction to Extension Fields 265
30 Vector Spaces 274
31 Algebraic Extensions 283
32 Geometric Constructions 293
33 Finite Fields 300
Ⅶ ADVANCED GROUP THEORY 307
34 Isomorphism Theorems 307
35 Series of Groups 311
36 Sylow Theorems 321
37 Applications of the Sylow Theory 327
38 Free Abelian Groups 333
39 Free Groups 341
40 Group Presentations 346
Ⅷ GROUPS IN TOPOLOGY 355
41 Simplicial Complexes and Homology Groups 355
42 Computations of Homology Groups 363
43 More Homology Computations and Applications 371
44 Homological Algebra 379
Ⅸ FACTORIZATION 389
45 Unique Factorization Domains 389
46 Euclidean Domains 401
47 Gaussian Integers and Multiplicative Norms 407
Ⅹ AUTOMORPHISMS AND GALOIS THEORY 415
48 Automorphisms of Fields 415
49 The Isomorphism Extension Theorem 424
50 Splitting Fields 431
51 Separable Extensions 436
52 Totally Inseparable Extensions 444
53 Galois Theory 448
54 Illustrations of Galois Theory 457
55 Cyclotomic Extensions 464
56 Insolvability of the Quintic 470
Appendix:Matrix Algebra 477
Bibliography 483
Notations 487
Answers to Odd-Numbered Exercises Not Asking for Definitions or Proofs 491
Index 513