FINITE COMMUTATIVE RINGS AND THEIR APPLICATIONS Gilberto Bini1 and Flaminio Flamini2 April 24, 2015 1University of Michigan, Dpt. of Mathematics, 525 East University Ave., Ann Arbor, MI, 48109, U.S.A., e-mail address: [email protected] 2Terza Universita’ di Roma ”Roma Tre”, Dip. di Matematica - Largo S. L. Murialdo, 1 - 00146 Roma, Italy, e-mail address: [email protected] Preface This book is a concrete and self-contained introduction to finite com- mutative local rings, focusing in particular on Galois and Quasi-Galois rings. Finite commutative ring theory is a fast-developing subject and has recently been seen to have important applications in theoretical areas like Combinatorics, Finite Geometries and the Analysis of Algorithms. Moreover, in the last twenty years, there has been a growing interest in application of commutative rings to Algebraic Cryptography and Coding Theory. In fact, several codes over finite fields, which are widely used in Information and Communication Theory, have been investigated as images of codes over Galois rings (especially over the ring of integers modulo 4). On the one side, applied mathematical research has mo- tivated a more systematic analysis of Finite Commutative Algebra; on the other side, pure Mathematics has offered innovative tools in Coding Theory. Therefore, this book aims to answer a need for introductory references in this evolving area from both perspectives. For this purpose, the reader is provided with an active and practical approach to the study of the purely algebraic structure and properties of finite commutative rings (in particular, Galois rings) as well as to their applications in Coding Theory. The Commutative Algebra set-up has been realized by the second author, whereas the Coding Theory point of view has been treated by the first author. This work is not intended as an exhaustive survey of all topics of ei- ther Finite Commutative Algebra or Coding Theory over finite rings. Mc Donald’s classical reference (see [56]) offers a more theoretical approach to the algebraic point of view of the subject. MacWilliams’ and Sloane’s book or van Lint’s book (see [53] and [69], respectively) - just to mention a few - are standard references for codes over finite fields, whereas [62] collects some of the latest articles concerning codes over Galois rings. This text could be appropriately used as a university course book i ii or for independent reading by students possessing some familiarity with basic algebraic topics, such as Group Theory, Commutative Rings, Finite Fields and Galois Theory. It should also be of great interest to engineers who have to deal in depth with Galois rings. Thus the first chapters can be viewed as a brief summary of basic definitions and results in Commutative Algebra. The reader is referred to a sufficiently detailed bibliography in order to avoid tedious repetitions of some too technical proofs. Together with Hensel’s lemma, the notion of regular polynomial is the fundamental tool of the entire work. Furthermore, in the chapters related to the separable extension theory of local rings, the crucial definitions of unramified extension of such rings and of the splitting ring of a regular polynomial are given. These extend the classical results of the Galois theory of finite fields to finite local rings. Chapter 6 is the core of the book, in which all results from previous chapters are used for the study of Galois rings and another class of finite local rings, Quasi-Galois rings. Moreover, an entire section is devoted to recalling some classical approaches to the theory of Galois rings. In Chapter 7 we briefly recall some standard definitions and results on codes over finite fields, which are necessary tools to discuss the formal duality between Kerdock and Preparata codes, one of the most intrigu- ing research topics in this area. In the last chapter, we deal with the explanation of this formal duality by using codes over finite rings. These two chapters are intended to point out the basic difference between codes over fields and over rings. We have tried to be as rigorous and accurate as possible, especially in proving the fundamental statements, at the same time keeping the examples lively and informal, since they may just be the key to the clarification of certain results. We would like to express our gratitude to everyone who helped and encouraged us throughout our years of study. Above all Prof. M.J. de Resmini, who has been a constant guide and without whom this work would never have come to life. We are indebted to Prof. Dr. D. Jungnickel for his precious and indispensable advice. We wish to thank our colleagues and friends for their support during the preparation of this book. Our deepest gratitude goes to our families. The second author would also like to thank his wife for her constant encouragement. Contents Preface ii 1 NOTIONS IN RING THEORY 1 1.1 BasicDefinitions ....................... 1 1.2 Prime and Maximal Ideals . 3 1.3 Euclidean Domains, P.I.D.’s and U.F.D.’s . 9 1.4 Factorization in Zpn [x]..................... 19 2 FINITE FIELD STRUCTURE 27 2.1 Basic Properties . 27 2.2 Characterization of Finite Fields . 29 2.3 Galois Field Automorphisms . 32 3 FINITE COMMUTATIVE RINGS 37 3.1 Finite Commutative Ring Structure . 37 3.2 Regular Polynomials in the Ring R[x] . 44 3.3 R-algebra Automorphisms of R[x] . 51 3.4 Factorization in R[x]..................... 53 4 SEPARABLE EXTENSIONS 59 4.1 Separable Field Extensions . 59 4.2 Extensions of Rings . 63 4.3 Separable extensions of local rings . 65 5 GALOIS THEORY FOR LOCAL RINGS 69 5.1 Basic Facts . 69 5.2 Examples. Splitting Rings . 73 6 GALOIS AND QUASI-GALOIS RINGS 79 6.1 Classical Constructions . 80 6.2 Galois Ring Properties . 90 iii iv CONTENTS 6.3 Structure Theorems . 103 6.4 Quasi-Galois Rings . 105 7 CODES OVER FINITE FIELDS 117 7.1 Basic properties . 117 7.2 Some families of q-ary codes . 118 7.2.1 Linear Codes . 118 7.2.2 Hamming codes . 119 7.2.3 Cyclic codes . 120 7.2.4 Reed-Muller codes . 124 7.3 Duality between codes . 126 7.4 Some families of nonlinear q-ary codes . 130 7.4.1 Binary Kerdock codes . 130 7.4.2 Kerdock sets . 130 7.4.3 Properties of binary Kerdock codes . 134 7.4.4 Classical Preparata codes . 136 7.4.5 Basic properties . 136 7.4.6 Preparata codes and Hamming codes . 137 8 CODES OVER GALOIS RINGS 141 8.1 Basic properties . 141 8.1.1 Linear codes over Zpn . 142 8.1.2 Reed-Muller codes over Zpn . 143 8.1.3 Cyclic codes over Zpn . 144 8.1.4 Hamming codes over Zpn . 147 8.2 Linear quaternary codes . 148 8.3 Kerdock and Preparata codes revisited . 154 Bibliography 163 Index 168 Chapter 1 FUNDAMENTAL NOTIONS IN RING THEORY We want to start by recalling some elementary topics in ring theory; we basically focus on local rings, since Galois rings, the ”main subject” of our work, are a particular class of such rings. We will review some definitions and provide clarifying examples. This is useful for the sake of establishing a common language, fixing, once and for all, notation such as would appear in many undergraduate Algebra texts whose contents we assume the reader is familiar with. 1.1 Basic Definitions From now on, by a ring we always mean a commutative ring with identity, unless explicitly stated. Let R be a ring. We recall that R is an integral domain if it contains no non-trivial zero-divisors. An element x R is ∈ nilpotent if xn = 0, for some positive integer n. So, a nilpotent element is a zero-divisor in R (provided R is not the trivial ring, i.e. R = 0), but the converse is not generally true. An invertible element (unit) x in R is an element for which there exists a y in R such that xy = 1, 1 being the multiplicative identity of R. The element y is uniquely determined by x and will be denoted by x−1. The subset U(R) := x R y R s.t. xy = yx = 1 { ∈ | ∃ ∈ } of R is a multiplicative group (with respect to the multiplication in R) and its elements are called the units of R. A ring R is a field if every 1 2 CHAPTER 1. NOTIONS IN RING THEORY non-zero element is a unit, i.e. U(R) = R∗ = R 0 . \{ } One of the most familiar examples of a (commutative and with iden- tity) ring is the ring of integers, denoted by Z, which trivially is an integral domain, but not a field; in fact, U(Z) = 1, 1 is isomorphic { − } to the cyclic group of order two, i.e. C =< x x2 = 1 >. If we consider 2 | the ring of the residues modulo m, for a fixed positive integer m, denoted by Zm= Z/mZ, we have a completely different situation. Proposition 1.1.1 Zm is an integral domain if and only if m is a prime. Proof: Left to the reader. ✷ More precisely, if m is a prime the structure of this ring is richer. Proposition 1.1.2 Assume m is a prime; then, given a Z , a = 0, ∈ m 6 there exists an element b such that ab = 1, i.e. Zm is a field. Proof: If m is a prime and a = 0, then m does not divide a in Z; 6 therefore g.c.d.(m, a) = (m, a) = 1. By the Euclidean algorithm, there exist integers r, b such that rm + ba = 1, thus ab = ba = 1 in Zm.