The smallest matrix of given period and primitive roots of unity

Abstract

A nonsingular matrix A has period n if A n = I but A k ≠ I for 0 < k < n. We investigate the number r K ( n), which is the smallest r such that there is an r × r matrix, with entries in the field K, that has period n. We compute this number as a function of the common degree θ K ( j) of the irreducible factors of the cyclotomic polynomial c j ( x). Thus, we are led to an investigation of roots of unity in order to better understand the function θ.