Table of Contents
1 Sets and Integers 1
1.1 Sets and Maps 1
1.2 The Factorization of Integers 7
1.3 Equivalence Relation and Partition 15
1.4 Exercises 18
2 Groups 21
2.1 The Concept of a Group and Examples 21
2.2 Subgroups and Cosets 31
2.3 Cyclic Groups 38
2.4 Exercises 45
3 Fields and Rings 49
3.1 Fields 49
3.2 The Characteristic of a Field 58
3.3 Rings and Integral Domains 64
3.4 Field of Fractions of an Integral Domain 68
3.5 Divisibility in a Ring 70
3.6 Exercises 72
4 Polynomials 75
4.1 Polynomial Rings 75
4.2 Division Algorithm 80
4.3 Euclidean Algorithm 83
4.4 Unique Factorization of Polynomials 93
4.5 Exercises 99
5 Residue Class Rings 101
5.1 Residue Class Rings 101
5.2 Examples 106
5.3 Residue Class Fields 108
5.4 More Examples 111
5.5 Exercises 114
6 Structure of Finite Fields 115
6.1 The Multiplicative Group of a Finite Field 115
6.2 The Number of Elements in a Finite Field 120
6.3 Existence of Finite Field with pn Elements 122
6.4 Uniqueness of Finite Field with pn Elements 127
6.5 Subfields of Finite Fields 128
6.6 A Distinction between Finite Fields of Characteristic 2 and Not 2 130
6.7 Exercises 133
7 Further Properties of Finite Fields 137
7.1 Automorphisms 137
7.2 Characteristic Polynomials and Minimal Polynomials 140
7.3 Primitive Polynomials 145
7.4 Trace and Norm 149
7.5 Quadratic Equations 156
7.6 Exercises. 158
8 Bases 161
8.1 Bases and Polynomial Bases 161
8.2 Dual Bases 166
8.3 Self-dual Bases 173
8.4 Normal Bases 180
8.5 Optimal Normal Bases 193
8.6 Exercises 206
9 Factoring Polynomials over Finite Fields 209
9.1 Factoring Polynomials over Finite Fields 209
9.2 Factorization of xn - 1 220
9.3 Cyclotomic Polynomials 224
9.4 The Period of a Polynomial 228
9.5 Exercises 235
10 Irreducible Polynomials over Finite Fields 237
10.1 On the Determination of Irreducible Polynomials 237
10.2 Irreducibility Criterion of Binomials 239
10.3 Some Irreducible Trinomials 243
10.4 Compositions of Polynomials 249
10.5 Recursive Constructions 255
10.6 Composed Product and Sum of Polynomials 259
10.7 Irreducible Polynomials of Any Degree 263
10.8 Exercises 265
11 Quadratic Forms over Finite Fields 269
11.1 Quadratic Forms over Finite Fields of Characteristic not 2 269
11.2 Alternate Forms over Finite Fields 278
11.3 Quadratic Forms over Finite Fields of Characteristic 2 282
11.4 Exercises 293
12 More Group Theory and Ring Theory 295
12.1 Homomorphisms of Groups, Normal Subgroups and Factor Groups 295
12.2 Direct Product Decomposition of Groups 303
12.3 Some Ring Theory 308
12.4 Modules 316
12.5 Exercises 327
13 Hensel's Lemma and Hensel Lift 329
13.1 The Polynomial Ring Zps[x] 329
13.2 Hensel's Lemma 332
13.3 Factorization of Monic Polynomials in Zps[x] 334
13.4 Basic Irreducible Polynomials and Hensel Lift 336
13.5 Exercises 340
14 Galois Rings 341
14.1 Examples of Galois Rings 341
14.2 Structure of Galois Rings 345
14.3 The p-adic Representation 349
14.4 The Group of Units of a Galois Ring 352
14.5 Extension of Galois Rings 356
14.6 Automorphisms of Galois Rings 361
14.7 Generalized Trace and Norm 365
14.8 Exercises 366
Bibliography 369
Index 373
