상세정보

This text covers the traditional approach of groups, rings, fields with the integration of computing and applications found in areas such as coding theory and cryptography. Applied examples are used to aid in the motivation of learning to prove theorems and propositions. The nature of exercises in this text range over several categories including computational, conceptual and theoretical. These exercises and problems allow the exploration of new results and theory. The flexible organization can be used in many different ways to emphasize theory or applications. It includes features and in text learning aids, applications within every chapter, quantity and quality of examples and exercises, supplementary topics, balance of theory and mathematics, historical notes, and computer science projects.

정보제공 :

CONTENTS
Preface = ⅶ
0 Preliminaries = 1
  0.1 A Short Note on Proofs = 1
  0.2 Sets and Equivalence Relations = 4
1 The Integers = 23
  1.1 Mathematical Induction = 23
  1.2 The Division Algorithm = 27
2 Groups = 37
  2.1 The Integers mod n and Symmetries = 37
  2.2 Definitions and Examples = 42
  2.3 Subgroups = 48
3 Cyclic Groups = 59
  3.1 Cyclic Subgroups = 59
  3.2 The Group C* = 63
  3.3 The Method of Repeated Squares = 68
4 Permutation Groups = 76
  4.1 Definitions and Notation = 77
  4.2 The Dihedral Groups = 85
5 Cosets and Lagrange's Theorem = 94
  5.1 Cosets = 94
  5.2 Lagrange's Theorem = 97
  5.3 Fermat's and Euler's Theorems = 99
6 Introduction to Cryptography = 103
  6.1 Private Key Cryptography = 104
  6.2 Public Key Cryptography = 107
7 Algebraic Coding Theory = 115
  7.1 Error-Detecting and Correcting Codes = 115
  7.2 Linear Codes = 124
  7.3 Parity-Check and Generator Matrices = 128
  7.4 Efficient Decoding = 135
8 Isomorphisms = 145
  8.1 Definition and Examples = 145
  8.2 Direct Products = 150
9 Homomorphisms and Factor Groups = 160
  9.1 Factor Groups and Normal Subgroups = 160
  9.2 Group Homomorphisms = 163
  9.3 The Isomorphism Theorems = 170
10 Matrix Groups and Symmetry = 179
  10.1 Matrix Groups = 179
  10.2 Symmetry = 188
11 The Structure of Groups = 200
  11.1 Finite Abelian Groups = 200
  11.2 Solvable Groups = 205
12 Group Actions = 213
  12.1 Groups Acting on Sets = 213
  12.2 The Class Equation = 217
  12.3 Burnside's Counting Theorem = 219
13 The Sylow Theorems = 231
  13.1 The Sylow Theorems = 231
  13.2 Examples and Applications = 235
14 Rings = 243
  14.1 Rings = 243
  14.2 Integral Domains and Fields = 248
  14.3 Ring Homomorphisms and Ideals = 250
  14.4 Maximal and Prime Ideals = 255
  14.5 An Application to Software Design = 257
15 Polynomials = 268
  15.1 Polynomial Rings = 269
  15.2 The Division Algorithm = 273
  15.3 Irreducible Polynomials = 277
16 Integral Domains = 289
  16.1 Fields of Fractions = 289
  16.2 Factorization in Integral Domains = 294
17 Lattices and Boolean Algebras = 307
  17.1 Lattices = 307
  17.2 Boolean Algebras = 312
  17.3 The Algebra of Electrical Circuits = 318
18 Vector Spaces = 326
  18.1 Definitions and Examples = 326
  18.2 Subspaces = 328
  18.3 Linear Independence = 329
19 Fields = 336
  19.1 Extension Fields = 336
  19.2 Splitting Fields = 348
  19.3 Geometric Constructions = 351
20 Finite Fields = 360
  20.1 Structure of a Finite Field = 360
  20.2 Polynomial Codes = 365
21 Galois Theory = 379
  21.1 Field Automorphisms = 379
  21.2 The Fundamental Theorem = 385
  21.3 Applications = 393
Notation = 402
Hints and Solutions = 406