Modern Algebra: An Introduction, 6th Edition

by Durbin, John R.

• ISBN13: 9780470384435
• ISBN10: 0470384433
• Edition: 6th
• Format: Hardcover
• Copyright: 2008-12-31
• Publisher: Wiley

Summary

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.

Author Biography

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.

Table of Contents

Introduction	p. 1
Mappings and Operations	p. 9
Mappings	p. 9
Composition. Invertible Mappings	p. 15
Operations	p. 19
Composition as an Operation	p. 25
Introduction to Groups	p. 30
Definition and Examples	p. 30
Permutations	p. 34
Subgroups	p. 41
Groups and Symmetry	p. 47
Equivalence. Congruence. Divisibility	p. 52
Equivalence Relations	p. 52
Congruence. The Division Algorithm	p. 57
Integers Modulo n	p. 61
Greatest Common Divisors. The Euclidean Algorithm	p. 65
Factorization. Euler's Phi-Function	p. 70
Groups	p. 75
Elementary Properties	p. 75
Generators. Direct Products	p. 81
Cosets	p. 85
Lagrange's Theorem. Cyclic Groups	p. 88
Isomorphism	p. 93
More on Isomorphism	p. 98
Cayley's Theorem	p. 102
RSA Algorithm	p. 105
Group Homomorphisms	p. 106
Homomorphisms of Groups. Kernels	p. 106
Quotient Groups	p. 110
The Fundamental Homomorphism Theorem	p. 114
Introduction to Rings	p. 120
Definition and Examples	p. 120
Integral Domains. Subrings	p. 125
Fields	p. 128
Isomorphism. Characteristic	p. 131
The Familiar Number Systems	p. 137
Ordered Integral Domains	p. 137
The Integers	p. 140
Field of Quotients. The Field of Rational Numbers	p. 142
Ordered Fields. The Field of Real Numbers	p. 146
The Field of Complex Numbers	p. 149
Complex Roots of Unity	p. 154
Polynomials	p. 160
Definition and Elementary Properties	p. 160
Appendix to Section 34	p. 162
The Division Algorithm	p. 165
Factorization of Polynomials	p. 169
Unique Factorization Domains	p. 173
Quotient Rings	p. 178
Homomorphisms of Rings. Ideals	p. 178
Quotient Rings	p. 182
Quotient Rings of F[X]	p. 184
Factorization and Ideals	p. 187
Galois Theory: Overview	p. 193
Simple Extensions. Degree	p. 194
Roots of Polynomials	p. 198
Fundamental Theorem: Introduction	p. 203
Galois Theory	p. 207
Algebraic Extensions	p. 207
Splitting Fields. Galois Groups	p. 210
Separability and Normality	p. 214
Fundamental Theorem of Galois Theory	p. 218
Solvability by Radicals	p. 219
Finite Fields	p. 223
Geometric Constructions	p. 229
Three Famous Problems	p. 229
Constructible Numbers	p. 233
Impossible Constructions	p. 234
Solvable and Alternating Groups	p. 237
Isomorphism Theorems and Solvable Groups	p. 237
Alternating Groups	p. 240
Applications of Permutation Groups	p. 243
Groups Acting on Sets	p. 243
Burnside's Counting Theorem	p. 247
Sylow's Theorem	p. 252
Symmetry	p. 256
Finite Symmetry Groups	p. 256
Infinite Two-Dimensional Symmetry Groups	p. 263
On Crystallographic Groups	p. 267
The Euclidean Group	p. 274
Lattices and Boolean Algebras	p. 279
Partially Ordered Sets	p. 279
Lattices	p. 283
Boolean Algebras	p. 287
Finite Boolean Algebras	p. 291
Sets	p. 296
Proofs	p. 299
Mathematical Induction	p. 304
Linear Algebra	p. 307
Solutions to Selected Problems	p. 312
Photo Credit List	p. 326
Index of Notation	p. 327
Index	p. 330
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