Introduction to Abstract Algebra

by Smith; Jonathan D.H.

• ISBN13: 9781420063714
• ISBN10: 1420063715
• Format: Hardcover
• Copyright: 2008-08-20
• Publisher: Chapman & Hall/

Summary

"Taking a slightly different approach from similar texts, Introduction to Abstract Algebra presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps readers fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles." "The book begins with a quick introduction to algebra, providing a foundation to exhibit irrational numbers and gain an appreciation of cryptography. It then covers all the basics of abstract algebra, including groups, homomorphisms, rings, and fields. The final chapters deal with more advanced topics, such as factorization, modules, group actions, and quasigroups. Study projects at the end of each chapter encompass a range of applications, offering additional opportunities for research."--BOOK JACKET.

Table of Contents

Numbers	p. 1
Ordering numbers	p. 1
The Well-Ordering Principle	p. 3
Divisibility	p. 5
The Division Algorithm	p. 6
Greatest common divisors	p. 9
The Euclidean Algorithm	p. 10
Primes and irreducibles	p. 13
The Fundamental Theorem of Arithmetic
Exercises	p. 17
Study projects	p. 22
Notes	p. 23
Functions	p. 25
Specifying functions	p. 25
Composite functions	p. 27
Linear functions	p. 28
Semigroups of functions	p. 29
Injectivity and surjectivity	p. 31
Isomorphisms	p. 34
Groups of permutations	p. 36
Exercises	p. 39
Study projects	p. 43
Notes	p. 46
Summary	p. 47
Equivalence	p. 49
Kernel and equivalence relations	p. 49
Equivalence classes	p. 51
Rational numbers	p. 53
The First Isomorphism Theorem for Sets	p. 56
Modular arithmetic	p. 58
Exercises	p. 61
Study projects	p. 63
Notes	p. 66
Groups and Monoids	p. 67
Semigroups	p. 67
Monoids	p. 69
Groups	p. 71
Componentwise structure	p. 73
Powers	p. 77
Submonoids and subgroups	p. 78
Cosets	p. 82
Multiplication tables	p. 84
Exercises	p. 87
Study projects	p. 91
Notes	p. 94
Homomorphisms	p. 95
Homomorphisms	p. 95
Normal subgroups	p. 98
Quotients	p. 101
The First Isomorphism Theorem for Groups	p. 104
The Law of Exponents	p. 106
Cayley's Theorem	p. 109
Exercises	p. 112
Study projects	p. 116
Notes	p. 125
Rings	p. 127
Rings	p. 127
Distributivity	p. 131
Subrings	p. 133
Ring homomorphisms	p. 135
Ideals	p. 137
Quotient rings	p. 139
Polynomial rings	p. 140
Substitution	p. 145
Exercises	p. 147
Study projects	p. 151
Notes	p. 156
Fields	p. 157
Integral domains	p. 157
Degrees	p. 160
Fields	p. 162
Polynomials over fields	p. 164
Principal ideal domains	p. 167
Irreducible polynomials	p. 170
Lagrange interpolation	p. 173
Fields of fractions	p. 175
Exercises	p. 178
Study projects	p. 182
Notes	p. 184
Factorization	p. 185
Factorization in integral domains	p. 185
Noetherian domains	p. 188
Unique factorization domains	p. 190
Roots of polynomials	p. 193
Splitting fields	p. 196
Uniqueness of splitting fields	p. 198
Structure of finite fields	p. 202
Galois fields	p. 204
Exercises	p. 206
Study projects	p. 210
Notes	p. 213
Modules	p. 215
Endomorphisms	p. 215
Representing a ring	p. 219
Modules	p. 220
Submodules	p. 223
Direct sums	p. 227
Free modules	p. 231
Vector spaces	p. 235
Abelian groups	p. 240
Exercises	p. 243
Study projects	p. 248
Notes	p. 251
Group Actions	p. 253
Actions	p. 253
Orbits	p. 256
Transitive actions	p. 258
Fixed points	p. 262
Faithful actions	p. 265
Cores	p. 267
Alternating groups	p. 270
Sylow Theorems	p. 273
Exercises	p. 277
Study projects	p. 283
Notes	p. 286
Quasigroups	p. 287
Quasigroups	p. 287
Latin squares	p. 289
Division	p. 293
Quasigroup homomorphisms	p. 297
Quasigroup homotopies	p. 301
Principal isotopy	p. 304
Loops	p. 306
Exercises	p. 311
Study projects	p. 315
Notes	p. 318
Index	p. 319
Table of Contents provided by Ingram. All Rights Reserved.