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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