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- 出版年份：1997
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1 Preliminaries 1
1.1 A Short Note on Proofs 1
1.2 Sets and Equivalence Relations 4
2 The Integers 22
2.1 Mathematical 1nduction 22
2.2 The Division Algorithm 26
3 Groups 35
3.1 Integer Equivalence Classes and Symmetries 35
3.2 Definitions and Examples 40
3.3 Subgroups 47
4 Cyclic Groups 57
4.1 Cyclic Subgroups 57
4.2 Multiplicative Group of Complex Numbers 61
4.3 The Method of Repeated Squares 66
5 Permutation Groups 74
5.1 Definitions and Notation 75
5.2 Dihedral Groups 83
6 Cosets and Lagrange’s Theorem 92
6.1 Cosets 92
6.2 Lagrange’s Theorem 95
6.3 Fermat’s and Euler’s Theorems 97
7 Introduction to Cryptography 100
7.1 Private Key Cryptography 101
7.2 Public Key Cryptography 104
8 Algebraic Coding Theory 111
8.1 Error-Detecting and Correcting Codes 111
8.2 Linear Codes 120
8.3 Parity-Check and Generator Matrices 124
8.4 Efficient Decoding 131
9 Isomorphisms 140
9.1 Definition and Examples 140
9.2 Direct Products 145
10 Normal Subgroups and Factor Groups 154
10.1 Factor Groups and Normal Subgroups 154
10.2 The Simplicity of the Alternating Group 157
11 Homomorphisms 164
11.1 Group Homomorphisms 164
11.2 The Isomorphism Theorems 167
12 Matrix Groups and Symmetry 174
12.1 Matrix Groups 174
12.2 Symmetry 183
13 The Structure of Groups 195
13.1 Finite Abelian Groups 195
13.2 Solvable Groups 200
14 Group Actions 208
14.1 Groups Acting on Sets 208
14.2 The Class Equation 212
14.3 Burnside’s Counting Theorem 214
15 The Sylow Theorems 226
15.1 The Sylow Theorems 226
15.2 Examples and Applications 230
16 Rings 238
16.1 Rings 238
16.2 Integral Domains and Fields 243
16.3 Ring Homomorphisms and Ideals 245
16.4 Maximal and Prime Ideals 249
16.5 An Application to Software Design 252
17 Polynomials 262
17.1 Polynomial Rings 263
17.2 The Division Algorithm 267
17.3 Irreducible Polynomials 271
18 Integral Domains 282
18.1 Fields of Fractions 282
18.2 Factorization in Integral Domains 286
19 Lattices and Boolean Algebras 300
19.1 Lattices 300
19.2 Boolean Algebras 305
19.3 The Algebra of Electrical Circuits 311
20 Vector Spaces 318
20.1 Definitions and Examples 318
20.2 Subspaces 320
20.3 Linear Independence 321
21 Fields 328
21.1 Extension Fields 328
21.2 Splitting Fields 339
21.3 Geometric Constructions 342
22 Finite Fields 352
22.1 Structure of a Finite Field 352
22.2 Polynomial Codes 357
23 Galois Theory 370
23.1 Field Automorphisms 370
23.2 The Fundamental Theorem 376
23.3 Applications 384
Hints and Solutions 393
GNU Free Documentation License 408
Notation 416
Index 420