Characterizations of the $d$th-power residue matrices over finite fields

Abstract

In a recent paper of the author with D. Dummit and H. Kisilevsky, we constructed a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of Q, and proved a simple criterion characterizing such matrices. We then analyzed the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of Q( √ −3) and Q(i), respectively. In this paper, the goal is to construct and study the finite-field analogues of these residue matrices, the "dth-power residue matrices", using the general dth-power residue symbol over a finite field.We remark here (and will justify later) that the resulting class of matrices is independent of the isomorphism ϕ: any other isomorphism will produce the same class of matrices.Definition. Let d be a positive integer. A "cyclotomic sign matrix of dth roots of unity" is an n × n matrix whose diagonal entries are all 0 and whose off-diagonal entries are all complex dth roots of unity.With the correct class of matrices in hand, we can now define the dth-power residue matrices.