Finite Fields and Galois Rings

386 Pages·2012·22.633 MB·English

by  Zhe-Xian Wan

FINITE FIELDS
AND
GALOIS RINGS

FINITE FIELDS
AND
GALOIS RINGS
Zhe-Xian Wa n
Chinese Academy of Scieeeee, China
\‘g—‘y World Scientific
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FINITE FIELDS AND GALOIS RINGS
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Preface
The present book is a revision and addendum ofa former one entitled Lec-
tures onFiniteFields and GaloisRings, whichwillbereferredas “Lectures”
in the following. The “Lectures” is based on a course on finite fields I gave
at Nankai University, Tianjin and a seminar on Galois rings I conducted at
SuzhouUniversity, Suzhou, bothin 2002. It hasbeen used as atextbookfor
students in mathematics as well as those majoring in computer science or
communication engineering. Ofcourse, it has also been used for self-study.
Thefirstfivechaptersand Chapter 12ofthe “Lectures” areprerequisites
for studying finite fields and Galois rings, respectively. They are prepared
for students who had no background on abstract algebra, for instance, stu-
dents in computer science or communication engineering. For those who
already took a course on abstract algebra these chapters may be skipped.
Chapter 6 to Chapter 11 of the “Lectures” are the main contents offinite
fields; they are: structuretheorems, automorphisms, norms andtraces, var-
ious bases, factoringpolynomials, constructingirreduciblepolynomials, and
quadratic forms over (or of) finite fields. Galois rings are treated in Chap—
ter 14, and Hensel’s Lemma and Hensel lift, which are needed in studying
Galois rings, are contained in Chapter 13.
In the present book many typos of the “Lectures” are corrected, some
changes are made for clarity, and many exercises are added. The most
significant additionstothe “Lectures” areasectiononoptimalnormalbasis
(i.e., section 8.5) and an expansion of the original section 12.4 so that a
simple proofofTheorem 8.34 is included.
PeopleareinterestedinfinitefieldsandGaloisrings, mainlybecausethey
have important applications in science and technology; for instances: shift
register sequences, algebraic coding theory, cryptography, design theory,
algebraic system theory, etc. However, in order to make this book not too
thick, these applications are not included.
In preparing the book I benefitted a lot from the existing literature on
vi PREFACE
finite fields and Galois rings; in particular, the encyclopedia of Lidl and
Niederreiter, the books of Jungnickel, McDonald, McEliece, and Menezes
et als.
The author is most gratefulto Professor Qinghu Hou ofNankai Univer—
sity and Drs. Yuping Deng, Ruoxia Du, Mei Fu, Jun-Wei Guo, Qiu—Min
Guo, Xia Jiang, De—Gang Liang, Chun—Lin Liu, Yan—Ping Mu, Yun Qin,
Chao Wang, Limin Yang, Ling-Ling Yang, Yiting Yang, Bao—Yin Zhang,
Jing—Yu Zhao who went through the tedious work of making a fair copy
of the book during their attending my lectures at Nankai University. The
author thanks Ms Jinling Chang who helped him with Latex problems and
the compiling ofthe index. Finally, the author is also indebted to Professor
Zhongming Tang of Suzhou University who read the whole text and gave
valuable comments.
Thanks are due to thosereaderswhotookthetrouble topoint out typos
ofthe “Lectures”.
Beijing, 2010 Zhe—Xian Wan
Contents
1 Sets and Integers 1
1.1 Sets and Maps .......................... 1
1.2 The Factorization ofIntegers . . . . .............. 7
I
1.3 Equivalence Relation and Partition ............... 15
1.4 Exercises ............................. 18
2 Groups 21
2.1 The Concept ofa 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 ofFractions ofan 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
Vii
viii CONTENTS
4.3 Euclidean Algorithm ....................... 83
4.4 Unique Factorization ofPolynomials .............. 93
4.5 Exercises ............................. 99
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
Structure of Finite Fields 115
6.1 The Multiplicative Group ofa Finite Field ........... 115
6.2 The Number ofElements in a Finite Field ........... 120
6.3 Existence ofFinite Field withp" Elements ........... 122
6.4 Uniqueness ofFinite Field with 1)" Elements .......... 127
6.5 Subfields ofFinite Fields ..................... 128
6.6 A Distinction between Finite Fields of Characteristic 2 and
Not 2 ............................... 130
6.7 Exercises ............................. 133
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
Bases . 161
8.1 Bases and Polynomial Bases ................... 161
8.2 Dual Bases ............................ 166
CONTENTS ix
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 ofx" — 1 ...................... 220
9.3 Cyclotomic Polynomials ..................... 224
9.4 The Period ofa Polynomial ................... 228
9.5 Exercises ............................. 235
10 Irreducible Polynomials over Finite Fields 237
10.1 On the Determination ofIrreducible Polynomials ....... 237
10.2 Irreducibility Criterion ofBinomials .............. 239
10.3 Some Irreducible Trinomials ................... 243
10.4 Compositions ofPolynomials .................. 249
10.5 Recursive Constructions ..................... 255
10.6 Composed Product and Sum ofPolynomials .......... 259
10.7 Irreducible Polynomials ofAny Degree ............. 263
10.8 Exercises ............................. 265
11 Quadratic Forms over Finite Fields - 269
11.1 Quadratic Forms over Finite Fields ofCharacteristic 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
X CONTENTS
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 s[:13] ................... 329
13.2 Hensel’s Lemma ......................... 332
13.3 Factorization ofMonic Polynomials in Z 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 ofUnits ofa Galois Ring .............. 352
14.5 Extension ofGalois Rings .................... 356
14.6 Automorphisms of Galois Rings ................. 361
14.7 Generalized Trace and Norm .................. 365
14.8 Exercises ............................. 366
Bibliography 369
Index 373