A First Course in Abstract Algebra, 7th Edition

1 Introduction and Examples............................................... 24
2 Binary Operations....................................................... 33
3 Isomorphic Binary Structures............................................ 41
4 Groups.................................................................. 49
5 Subgroups............................................................... 62
6 Cyclic Groups........................................................... 72
7 Generating Sets and Cayley Digraphs..................................... 81
8 Groups of Permutations.................................................. 88
9 Orbits, Cycles, and the Alternating Groups..............................100
10 Cosets and the Theorem of Lagrange.....................................109
11 Direct Products and Finitely Generated Abelian Groups..................117
12 Plane Isometries.......................................................127
13 Homomorphisms..........................................................138
14 Factor Groups..........................................................148
15 Factor-Group Computations and Simple Groups............................157
16 Group Action on a Set..................................................167
17 Applications of G-Sets to Counting.....................................174
18 Rings and Fields.......................................................180
19 Integral Domains.......................................................190
20 Fermat's and Euler's Theorems..........................................197
21 The Field of Quotients of an Integral Domain...........................203
22 Rings of Polynomials...................................................211
23 Factorization of Polynomials over a Field..............................222
24 Noncommutative Examples................................................233
25 Ordered Rings and Fields...............................................240
26 Homomorphisms and Factor Rings.........................................250
27 Prime and Maximal Ideals...............................................258
28 Grobner Bases for Ideals...............................................267
29 Introduction to Extension Fields.......................................278
30 Vector Spaces..........................................................287
31 Algebraic Extensions...................................................296
32 Geometric Constructions................................................306
33 Finite Fields..........................................................313
34 Isomorphism Theorems...................................................320
35 Series of Groups.......................................................324
36 Sylow Theorems.........................................................334
37 Applications of the Sylow Theory.......................................340
38 Free Abelian Groups....................................................346
39 Free Groups............................................................354
40 Group Presentations....................................................359
41 Simplicial Complexes and Homology Groups...............................368
42 Computations of Homology Groups........................................376
43 More Homology Computations and Applications............................384
44 Homological Algebra....................................................392
45 Unique Factorization Domains...........................................402
46 Euclidean Domains......................................................414
47 Gaussian Integers and Multiplicative Nonns.............................420
48 Automorphisms of Fields................................................428
49 The Isomorphism Extension Theorem......................................437
50 Splitting Fields.......................................................444
51 Separable Extensions...................................................449
52 Totally Inseparable Extensions.........................................457
53 Galois Theory..........................................................461
54 Illustrations of Galois Theory.........................................470
55 Cyclotomic Extensions..................................................477
56 Insolvability of the Quintic...........................................483
Appendix: Matrix Algebra..................................................490
Bibliography..............................................................496
Notations.................................................................500
Answers to Odd-Numbered Exercises Not Asking for Definitions or Proofs....504
Index.....................................................................526