Modern Algebra with Applications

Blending the theoretical with the practical in the instruction of modern algebra, Modern Algebra with Applications, Second Edition provides interesting and important applications of this subject-effectively holding your interest and creating a more seamless method of instruction. Filled with in-depth insights and over six hundred exercises of varying difficulty, this invaluable text can help anyone appreciate and understand this subject.

WILLIAM J. GILBERT, DPHil, is a professor in the Department of Pure Mathematics at the University of Waterloo, Ontario, Canada. He received his DPhil in mathematics from Oxford University in 1968. W. KEITH NICHOLSON, PHD, is a professor in the Department of Mathematics and Statistics at the University of Calgary, Alberta, Canada. He received his PhD in pure mathematics from the University of California at Santa Barbara in 1970.

Preface to the First Edition

ix

Preface to the Second Edition

xiii

List of Symbols

xv

1 Introduction

1

(6)

Classical Algebra,

1

(1)

Modern Algebra,

2

(1)

Binary Operations,

2

(2)

Algebraic Structures,

4

(1)

Extending Number Systems,

5

(2)

2 Boolean Algebras

7

(40)

Algebra of Sets,

7

(4)

Number of Elements in a Set,

11

(2)

Boolean Algebras,

13

(3)

Propositional Logic,

16

(3)

Switching Circuits,

19

(2)

Divisors,

21

(2)

Poseis and Lattices,

23

(3)

Normal Forms and Simplification of Circuits,

26

(10)

Transistor Gates,

36

(3)

Representation Theorem,

39

(2)

Exercises,

41

(6)

3 Groups

47

(29)

Groups and Symmetries,

48

(6)

Subgroups,

54

(2)

Cyclic Groups and Dihedral Groups,

56

(4)

Morphisms,

60

(3)

Permutation Groups,

63

(4)

Even and Odd Permutations,

67

(4)

Cayley's Representation Theorem,

71

(1)

Exercises,

71

(5)

4 Quotient Groups

76

(28)

Equivalence Relations,

76

(2)

Cosecs and Lagrange' s Theorem,

78

(4)

Normal Subgroups and Quotient Groups,

82

(4)

Morphism Theorem,

86

(5)

Direct Products,

91

(3)

Groups of Low Order,

94

(2)

Action of a Group on a Set,

96

(3)

Exercises,

99

(5)

5 Symmetry Groups in Three Dimensions

104

(20)

Translations and the Euclidean Group,

104

(3)

Matrix Groups,

107

(2)

Finite Groups in Two Dimensions,

109

(2)

Proper Rotations of Regular Solids,

111

(5)

Finite Rotation Groups in Three Dimensions,

116

(4)

Crystallographic Groups,

120

(1)

Exercises,

121

(3)

6 Pólya-Burnside Method of Enumeration

124

(13)

Burnside's Theorem,

124

(2)

Necklace Problems,

126

(2)

Coloring Polyhedra,

128

(2)

Counting Switching Circuits,

130

(4)

Exercises,

134

(3)

7 Monoids and Machines

137

(18)

Monoids and Semigroups,

137

(5)

Finite-State Machines,

142

(2)

Quotient Monoids and the Monoid of a Machine,

144

(5)

Exercises,

149

(6)

8 Rings and Fields

155

(25)

Rings,

155

(4)

Integral Domains and Fields,

159

(2)

Subrings and Morphisms of Rings,

161

(3)

New Rings from Old,

164

(6)

Field of Fractions,

170

(2)

Convolution Fractions,

172

(4)

Exercises,

176

(4)

9 Polynomial and Euclidean Rings

180

(24)

Euclidean Rings,

180

(4)

Euclidean Algorithm,

184

(3)

Unique Factorization,

187

(3)

Factoring Real and Complex Polynomials,

190

(2)

Factoring Rational and Integral Polynomials,

192

(3)

Factoring Polynomials over Finite Fields,

195

(2)

Linear Congruences and the Chinese Remainder Theorem,

197

(4)

Exercises,

201

(3)

10 Quotient Rings

204

(14)

Ideals and Quotient Rings,

204

(3)

Computations in Quotient Rings,

207

(2)

Morphism Theorem,

209

(1)

Quotient Polynomial Rings That Are Fields,

210

(4)

Exercises,

214

(4)

11 Field Extensions

218

(18)

Field Extensions,

218

(3)

Algebraic Numbers,

221

(4)

Galois Fields,

225

(3)

Primitive Elements,

228

(4)

Exercises,

232

(4)

12 Latin Squares

236

(15)

Latin Squares,

236

(2)

Orthogonal Latin Squares,

238

(4)

Finite Geometries,

242

(3)

Magic Squares,

245

(4)

Exercises,

249

(2)

13 Geometrical Constructions

251

(13)

Constructible Numbers,

251

(5)

Duplicating a Cube,

256

(1)

Trisecting an Angle,

257

(2)

Squaring the Circle,

259

(1)

Constructing Regular Polygons,

259

(1)

Nonconstructible Number of Degree 4,

260

(2)

Exercises,

262

(2)

14 Error-Correcting Codes

264

(29)

The Coding Problem,

266

(1)

Simple Codes,

267

(3)

Polynomial Representation,

270

(6)

Matrix Representation,

276

(4)

Error Correcting and Decoding,

280

(4)

BCH Codes,

284

(4)

Exercises,

288

(5)

Appendix 1: Proofs

293

(3)

Appendix 2: Integers

296

(10)

Bibliography and References

306

(3)

Answers to Odd-Numbered Exercises

309

(14)

Index

323

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Modern Algebra with Applications

Summary

Author Biography

Table of Contents

Supplemental Materials