《ABSTRACT ALGEBRA THEORY AND APPLICATIONS》

1Preliminaries1

1.1 A Short Note on Proofs1

1.2 Sets and Equivalence Relations4

2The Integers22

2.1 Mathematical 1nduction22

2.2 The Division Algorithm26

3Groups35

3.1 Integer Equivalence Classes and Symmetries35

3.2 Definitions and Examples40

3.3 Subgroups47

4Cyclic Groups57

4.1 Cyclic Subgroups57

4.2 Multiplicative Group of Complex Numbers61

4.3 The Method of Repeated Squares66

5Permutation Groups74

5.1 Definitions and Notation75

5.2 Dihedral Groups83

6Cosets and Lagrange’s Theorem92

6.1 Cosets92

6.2 Lagrange’s Theorem95

6.3 Fermat’s and Euler’s Theorems97

7Introduction to Cryptography100

7.1 Private Key Cryptography101

7.2 Public Key Cryptography104

8Algebraic Coding Theory111

8.1 Error-Detecting and Correcting Codes111

8.2 Linear Codes120

8.3 Parity-Check and Generator Matrices124

8.4 Efficient Decoding131

9Isomorphisms140

9.1 Definition and Examples140

9.2 Direct Products145

10 Normal Subgroups and Factor Groups154

10.1 Factor Groups and Normal Subgroups154

10.2 The Simplicity of the Alternating Group157

11 Homomorphisms164

11.1 Group Homomorphisms164

11.2 The Isomorphism Theorems167

12 Matrix Groups and Symmetry174

12.1 Matrix Groups174

12.2 Symmetry183

13 The Structure of Groups195

13.1 Finite Abelian Groups195

13.2 Solvable Groups200

14 Group Actions208

14.1 Groups Acting on Sets208

14.2 The Class Equation212

14.3 Burnside’s Counting Theorem214

15 The Sylow Theorems226

15.1 The Sylow Theorems226

15.2 Examples and Applications230

16 Rings238

16.1 Rings238

16.2 Integral Domains and Fields243

16.3 Ring Homomorphisms and Ideals245

16.4 Maximal and Prime Ideals249

16.5 An Application to Software Design252

17 Polynomials262

17.1 Polynomial Rings263

17.2 The Division Algorithm267

17.3 Irreducible Polynomials271

18 Integral Domains282

18.1 Fields of Fractions282

18.2 Factorization in Integral Domains286

19 Lattices and Boolean Algebras300

19.1 Lattices300

19.2 Boolean Algebras305

19.3 The Algebra of Electrical Circuits311

20 Vector Spaces318

20.1 Definitions and Examples318

20.2 Subspaces320

20.3 Linear Independence321

21 Fields328

21.1 Extension Fields328

21.2 Splitting Fields339

21.3 Geometric Constructions342

22 Finite Fields352

22.1 Structure of a Finite Field352

22.2 Polynomial Codes357

23 Galois Theory370

23.1 Field Automorphisms370

23.2 The Fundamental Theorem376

23.3 Applications384

Hints and Solutions393

GNU Free Documentation License408

Notation416

Index420