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

TL;DR

This paper extends the study of residue matrices by constructing and analyzing the $d$th-power residue matrices over finite fields, generalizing previous quadratic, cubic, and quartic residue matrix frameworks.

Contribution

It introduces the concept of $d$th-power residue matrices over finite fields and explores their properties, broadening the understanding of residue matrices beyond quadratic cases.

Findings

Constructed $d$th-power residue matrices over finite fields.

Provided criteria characterizing these matrices.

Analyzed their algebraic and combinatorial properties.

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 $\mathbb{Q}$, 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 $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively. In this paper, the goal is to construct and study the finite-field analogues of these residue matrices, the "$d$th-power residue matrices", using the general $d$th-power residue symbol over a finite field.