Modern Algebra with Applications

Description

Praise for the first edition ''This book is clearly written and presents a large number of examples illustrating the theory . . . there is no other book of comparable content available. Because of its detailed coverage of applications generally neglected in the literature, it is a desirable if not essential addition to undergraduate mathematics and computer science libraries.'' –CHOICE As a cornerstone of mathematical science, the importance of modern algebra and discrete structures to many areas of science and technology is apparent and growing–with extensive use in computing science, physics, chemistry, and data communications as well as in areas of mathematics such as combinatorics. 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. Incorporating the applications of modern algebra throughout its authoritative treatment of the subject, this book covers the full complement of group, ring, and field theory typically contained in a standard modern algebra course. Numerous examples are included in each chapter, and answers to odd-numbered exercises are appended in the back of the text. Chapter topics include:Boolean AlgebrasPolynomial and Euclidean RingsGroupsQuotient RingsQuotient GroupsField ExtensionsSymmetry Groups in Three DimensionsLatin SquaresPólya—Burnside Method of EnumerationGeometrical ConstructionsMonoids and MachinesError-Correcting CodesRings and Fields In addition to improvements in exposition, this fully updated Second Edition also contains new material on order of an element and cyclic groups, more details about the lattice of divisors of an integer, and new historical notes. Filled with in-depth insights and over 600 exercises of varying difficulty, Modern Algebra with Applications, Second Edition can help anyone appreciate and understand this subject. MODERN ALGEBRA WITH APPLICATIONS......Page 3 CONTENTS......Page 7 Preface to the First Edition......Page 11 Preface to the Second Edition......Page 15 List of Symbols......Page 17 Classical Algebra......Page 21 Binary Operations......Page 22 Algebraic Structures......Page 24 Extending Number Systems......Page 25 Algebra of Sets......Page 27 Number of Elements in a Set......Page 31 Boolean Algebras......Page 33 Propositional Logic......Page 36 Switching Circuits......Page 39 Divisors......Page 41 Posets and Lattices......Page 43 Normal Forms and Simplification of Circuits......Page 46 Transistor Gates......Page 56 Representation Theorem......Page 59 Exercises......Page 61 3 Groups......Page 67 Groups and Symmetries......Page 68 Subgroups......Page 74 Cyclic Groups and Dihedral Groups......Page 76 Morphisms......Page 80 Permutation Groups......Page 83 Even and Odd Permutations......Page 87 Exercises......Page 91 Equivalence Relations......Page 96 Cosets and Lagrange’s Theorem......Page 98 Normal Subgroups and Quotient Groups......Page 102 Morphism Theorem......Page 106 Direct Products......Page 111 Groups of Low Order......Page 114 Action of a Group on a Set......Page 116 Exercises......Page 120 Translations and the Euclidean Group......Page 124 Matrix Groups......Page 127 Finite Groups in Two Dimensions......Page 129 Proper Rotations of Regular Solids......Page 131 Finite Rotation Groups in Three Dimensions......Page 136 Crystallographic Groups......Page 140 Exercises......Page 141 Burnside’s Theorem......Page 144 Necklace Problems......Page 146 Coloring Polyhedra......Page 148 Counting Switching Circuits......Page 150 Exercises......Page 155 Monoids and Semigroups......Page 157 Finite-State Machines......Page 162 Quotient Monoids and the Monoid of a Machine......Page 164 Exercises......Page 169 Rings......Page 175 Integral Domains and Fields......Page 179 Subrings and Morphisms of Rings......Page 181 New Rings from Old......Page 184 Field of Fractions......Page 190 Convolution Fractions......Page 192 Exercises......Page 196 Euclidean Rings......Page 200 Euclidean Algorithm......Page 204 Unique Factorization......Page 207 Factoring Real and Complex Polynomials......Page 210 Factoring Rational and Integral Polynomials......Page 212 Factoring Polynomials over Finite Fields......Page 215 Linear Congruences and the Chinese Remainder Theorem......Page 217 Exercises......Page 221 Ideals and Quotient Rings......Page 224 Computations in Quotient Rings......Page 227 Morphism Theorem......Page 229 Quotient Polynomial Rings That Are Fields......Page 230 Exercises......Page 234 Field Extensions......Page 238 Algebraic Numbers......Page 241 Galois Fields......Page 245 Primitive Elements......Page 248 Exercises......Page 253 Latin Squares......Page 256 Orthogonal Latin Squares......Page 258 Finite Geometries......Page 262 Magic Squares......Page 265 Exercises......Page 269 Constructible Numbers......Page 271 Duplicating a Cube......Page 276 Trisecting an Angle......Page 277 Constructing Regular Polygons......Page 279 Nonconstructible Number of Degree 4......Page 280 Exercises......Page 282 14 Error-Correcting Codes......Page 284 The Coding Problem......Page 286 Simple Codes......Page 287 Polynomial Representation......Page 290 Matrix Representation......Page 296 Error Correcting and Decoding......Page 300 BCH Codes......Page 304 Exercises......Page 308 Appendix 1: Proofs......Page 313 Appendix 2: Integers......Page 316 Bibliography and References......Page 326 Answers to Odd-Numbered Exercises......Page 329 Index......Page 343