Modern Algebra

Dr. John R. Durbin is a professor of Mathematics at The University of Texas Austin. A native Kansan, he received B.A. and M.A. degrees from the University of Wichita (now Wichita State University), and a Ph.D. from the University of Kansas. He came to UT immediately thereafter. Professor Durbin has been active in faculty governance at the University for many years. He served as chair of the Faculty Senate, 1982-84 and 1991-92, and as Secretary of the General Faculty, 1975-76 and 1998-2003. In September of 2003 he received the University & Civitatis Award,in recognition of dedicated and meritorious service to the University above and beyond the regular expectations of teaching, research, and writing. He has received a Teaching Excellence Award from the College of Natural Sciences and an Outstanding Teaching Award from the Department of Mathematics.

Introduction 1

I. Mappings and Operations 9

1 Mappings 9

2 Composition. Invertible Mappings 15

3 Operations 19

4 Composition as an Operation 25

II. Introduction to Groups 30

5 Definition and Examples 30

6 Permutations 34

7 Subgroups 41

8 Groups and Symmetry 47

III. Equivalence. Congruence. Divisibility 52

9 Equivalence Relations 52

10 Congruence. The Division Algorithm 57

11 Integers Modulo n 61

12 Greatest Common Divisors. The Euclidean Algorithm 65

13 Factorization. Euler’s Phi-Function 70

IV. Groups 75

14 Elementary Properties 75

15 Generators. Direct Products 81

16 Cosets 85

17 Lagrange’s Theorem. Cyclic Groups 88

18 Isomorphism 93

19 More on Isomorphism 98

20 Cayley’s Theorem 102

Appendix: RSA Algorithm 105

V. Group Homomorphisms 106

21 Homomorphisms of Groups. Kernels 106

22 Quotient Groups 110

23 The Fundamental Homomorphism Theorem 114

VI. Introduction to Rings 120

24 Definition and Examples 120

25 Integral Domains. Subrings 125

26 Fields 128

27 Isomorphism. Characteristic 131

VII. The Familiar Number Systems 137

28 Ordered Integral Domains 137

29 The Integers 140

30 Field of Quotients. The Field of Rational Numbers 142

31 Ordered Fields. The Field of Real Numbers 146

32 The Field of Complex Numbers 149

33 Complex Roots of Unity 154

VIII. Polynomials 160

34 Definition and Elementary Properties 160

Appendix to Section 34 162

35 The Division Algorithm 165

36 Factorization of Polynomials 169

37 Unique Factorization Domains 173

IX. Quotient Rings 178

38 Homomorphisms of Rings. Ideals 178

39 Quotient Rings 182

40 Quotient Rings of F[X] 184

41 Factorization and Ideals 187

X. Galois Theory: Overview 193

42 Simple Extensions. Degree 194

43 Roots of Polynomials 198

44 Fundamental Theorem: Introduction 203

XI. Galois Theory 207

45 Algebraic Extensions 207

46 Splitting Fields. Galois Groups 210

47 Separability and Normality 214

48 Fundamental Theorem of Galois Theory 218

49 Solvability by Radicals 219

50 Finite Fields 223

XII. Geometric Constructions 229

51 Three Famous Problems 229

52 Constructible Numbers 233

53 Impossible Constructions 234

XIII. Solvable and Alternating Groups 237

54 Isomorphism Theorems and Solvable Groups 237

55 Alternating Groups 240

XIV. Applications of Permutation Groups 243

56 Groups Acting on Sets 243

57 Burnside’s Counting Theorem 247

58 Sylow’s Theorem 252

XV. Symmetry 256

59 Finite Symmetry Groups 256

60 Infinite Two-Dimensional Symmetry Groups 263

61 On Crystallographic Groups 267

62 The Euclidean Group 274

XVI. Lattices and Boolean Algebras 279

63 Partially Ordered Sets 279

64 Lattices 283

65 Boolean Algebras 287

66 Finite Boolean Algebras 291

A. Sets 296

B. Proofs 299

C. Mathematical Induction 304

D. Linear Algebra 307

E. Solutions to Selected Problems 312

Photo Credit List 326

Index of Notation 327

Index 330