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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