Finite Fields And Galois Rings

by Zhe-xian Wan

Overview

A large portion of the book can be used as a textbook for graduate and upper level undergraduate students in mathematics, communication engineering, computer science and other fields. The remaining part can be used as references for specialists. Explicit construction and computation of finite fields are emphasized. In particular, the construction of irreducible polynomials and normal basis of finite field is included. A detailed treatment of optimal normal basis and Galoi's rings is included. It is the first time that the galois rings are in book form.

Product Details

ISBN-13:	9789814366342
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	09/14/2011
Pages:	388
Product dimensions:	6.10(w) x 9.00(h) x 1.00(d)

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 Z_p^s[x] 329

13.2 Hensel's Lemma 332

13.3 Factorization of Monic Polynomials in Z_p^s[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