Applications Of Cubical Arrays In The Study Of Finite Semifields

Abstract

It is well known that any finite semifield, S, can be viewed as an n-dimensional vector space over a finite field or prime order, Fp, and that the multiplication in S defines and can be defined by an n x n x n cubical array of scalars, A. For any element a E S, the matrix, La, corresponding to left multiplication by a can be determined from A. In this paper we show that there exists a unique monic polynomial of minimal degree, f E Fp[x], such that f(a) = 0, and which divides the minimal polynomial of La. Furthermore, we show that some properties of f in Fp[x] correspond to properties of a in S. These results, in turn, help optimize a method we introduce which uses A to determine the automorphism group of S. We show that under certain conditions A can be inflated to define a new semifield, S[m], over the field Fpm , and that inflation preserves isotopism and isomorphism between inflated semifields. Finally, we apply our results to the 16-element semifields, and give algebraic constructions for each of these semifields for which no construction currently exists.