Finite fields and Galois rings

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Title:

Personal Author:

Wan, Zhexian

Physical Description:

x, 376 pages : illustrations ; 24 cm.

ISBN:

9789814366342

Subject Term:

Finite fields (Algebra)

Galois modules (Algebra)

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010280456	QA247.3 W36 2012	Open Access Book	Book	Searching... Unknown

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.

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

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