Modern algebra with applications / William J. Gilbert.

Format:

Book

Author/Creator:

Gilbert, William J., 1941-

Language:

English

Subjects (All):

Algebra, Abstract.

Physical Description:

xii, 348 pages : illustrations ; 24 cm

Place of Publication:

New York : Wiley, [1976]

Contents:

Classical Algebra 1

Modern Algebra 2

Binary Operations 2

Algebraic Structures 5

Extending Number Systems 6

2. Boolean Algebras 8

Algebra of Sets 8

Number of Elements in a Set 13

Boolean Algebras 15

Switching Circuits 23

Posets and Lattices 25

Normal Forms and Simplification of Circuits 28

Transistor Gates 39

Representation Theorem 42

3. Groups 51

Groups and Symmetries 52

Subgroups 58

Cyclic Groups and Dihedral Groups 60

Morphisms 64

Permutation Groups 67

Even and Odd Permutations 73

Cayley's Representation Theorem 77

4. Quotient Groups 82

Equivalence Relations 82

Cosets and Lagrange's Theorem 85

Normal Subgroups and Quotient Groups 88

Morphism Theorem 93

Direct Products 98

Groups of Low Order 102

Action of a Group on a Set 104

5. Symmetry Groups in Three Dimensions 112

Translations and the Euclidean Group 112

Matrix Groups 115

Finite Groups in Two Dimensions 117

Proper Rotations of Regular Solids 119

Finite Rotation Groups in Three Dimensions 124

Crystallographic Groups 129

6. Polya-Burnside Method of Enumeration 133

Burnside's Theorem 134

Necklace Problems 135

Coloring Polyhedra 138

Counting Switching Circuits 140

7. Monoids and Machines 147

Monoids and Semigroups 147

Finite-State Machines 152

Quotient Monoids and the Monoid of a Machine 155

8. Rings and Fields 166

Rings 166

Integral Domains and Fields 171

Subrings and Morphisms of Rings 173

New Rings from Old 176

Field of Fractions 182

Convolution Fractions 184

9. Polynomial and Euclidean Rings 193

Division Algorithm 193

Euclidean Algorithm 198

Unique Factorization 202

Factoring Real and Complex Polynomials 205

Factoring Rational and Integral Polynomials 207

Factoring Polynomials over Finite Fields 211

Linear Congruences and the Chinese Remainder Theorem 212

10. Quotient Rings 221

Ideals and Quotient Rings 221

Computations in Quotient Rings 225

Morphism Theorem 227

Quotient Polynomial Rings that are Fields 228

11. Field Extensions 235

Field Extensions 235

Algebraic Numbers 238

Galois Fields 243

Primitive Elements 246

12. Latin Squares 255

Latin Squares 255

Orthogonal Latin Squares 258

Finite Geometries 262

Magic Squares 266

13. Geometrical Constructions 272

Constructible Numbers 273

Duplicating the Cube 278

Trisecting an Angle 278

Squaring the Circle 281

Constructing Regular Polygons 281

A Nonconstructible Number of Degree Four 283

14. Error-Correcting Codes 286

The Coding Problem 287

Simple Codes 290

Polynomial Representation 293

Matrix Representation 299

Error Correcting and Decoding 304

BCH Codes 309.

Notes:

"A Wiley-Interscience publication."

Includes index.

Bibliography: pages 319-323.

ISBN:

0471298913

OCLC:

2331747