Modern Algebra: An Introduction, 6th Edition

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  • Edition: 6th
  • Format: Hardcover
  • Copyright: 2008-12-01
  • Publisher: Wiley

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Engineers and computer scientists who need a basic understanding of algebra will benefit from this accessible book. The sixth edition includes many carefully worked examples and proofs to guide them through abstract algebra successfully. It introduces the most important kinds of algebraic structures, and helps them improve their ability to understand and work with abstract ideas. New and revised exercise sets are integrated throughout the first four chapters. A more in-depth discussion is also included on Galois Theory. The first six chapters provide engineers and computer scientists with the core of the subject and then the book explores the concepts in more detail.

Table of Contents

Introductionp. 1
Mappings and Operationsp. 9
Mappingsp. 9
Composition. Invertible Mappingsp. 15
Operationsp. 19
Composition as an Operationp. 25
Introduction to Groupsp. 30
Definition and Examplesp. 30
Permutationsp. 34
Subgroupsp. 41
Groups and Symmetryp. 47
Equivalence. Congruence. Divisibilityp. 52
Equivalence Relationsp. 52
Congruence. The Division Algorithmp. 57
Integers Modulo np. 61
Greatest Common Divisors. The Euclidean Algorithmp. 65
Factorization. Euler's Phi-Functionp. 70
Groupsp. 75
Elementary Propertiesp. 75
Generators. Direct Productsp. 81
Cosetsp. 85
Lagrange's Theorem. Cyclic Groupsp. 88
Isomorphismp. 93
More on Isomorphismp. 98
Cayley's Theoremp. 102
RSA Algorithmp. 105
Group Homomorphismsp. 106
Homomorphisms of Groups. Kernelsp. 106
Quotient Groupsp. 110
The Fundamental Homomorphism Theoremp. 114
Introduction to Ringsp. 120
Definition and Examplesp. 120
Integral Domains. Subringsp. 125
Fieldsp. 128
Isomorphism. Characteristicp. 131
The Familiar Number Systemsp. 137
Ordered Integral Domainsp. 137
The Integersp. 140
Field of Quotients. The Field of Rational Numbersp. 142
Ordered Fields. The Field of Real Numbersp. 146
The Field of Complex Numbersp. 149
Complex Roots of Unityp. 154
Polynomialsp. 160
Definition and Elementary Propertiesp. 160
Appendix to Section 34p. 162
The Division Algorithmp. 165
Factorization of Polynomialsp. 169
Unique Factorization Domainsp. 173
Quotient Ringsp. 178
Homomorphisms of Rings. Idealsp. 178
Quotient Ringsp. 182
Quotient Rings of F[X]p. 184
Factorization and Idealsp. 187
Galois Theory: Overviewp. 193
Simple Extensions. Degreep. 194
Roots of Polynomialsp. 198
Fundamental Theorem: Introductionp. 203
Galois Theoryp. 207
Algebraic Extensionsp. 207
Splitting Fields. Galois Groupsp. 210
Separability and Normalityp. 214
Fundamental Theorem of Galois Theoryp. 218
Solvability by Radicalsp. 219
Finite Fieldsp. 223
Geometric Constructionsp. 229
Three Famous Problemsp. 229
Constructible Numbersp. 233
Impossible Constructionsp. 234
Solvable and Alternating Groupsp. 237
Isomorphism Theorems and Solvable Groupsp. 237
Alternating Groupsp. 240
Applications of Permutation Groupsp. 243
Groups Acting on Setsp. 243
Burnside's Counting Theoremp. 247
Sylow's Theoremp. 252
Symmetryp. 256
Finite Symmetry Groupsp. 256
Infinite Two-Dimensional Symmetry Groupsp. 263
On Crystallographic Groupsp. 267
The Euclidean Groupp. 274
Lattices and Boolean Algebrasp. 279
Partially Ordered Setsp. 279
Latticesp. 283
Boolean Algebrasp. 287
Finite Boolean Algebrasp. 291
Setsp. 296
Proofsp. 299
Mathematical Inductionp. 304
Linear Algebrap. 307
Solutions to Selected Problemsp. 312
Photo Credit Listp. 326
Index of Notationp. 327
Indexp. 330
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