On cyclotomic matrices involving Gauss sums over finite fields

TL;DR

This paper explores cyclotomic matrices involving Gauss sums over finite fields, providing explicit determinant formulas and extending classical results related to the Gamma function to finite field analogues.

Contribution

It introduces new determinant formulas for matrices constructed from Gauss sums over finite fields, generalizing classical cyclotomic matrix results.

Findings

Derived explicit determinant formulas for matrices of Gauss sums

Extended classical cyclotomic matrix results to finite fields

Identified conditions on prime powers affecting determinants

Abstract

Inspired by the works of L. Carlitz and Z.-W. Sun on cyclotomic matrices, in this paper, we investigate certain cyclotomic matrices involving Gauss sums over finite fields, which can be viewed as finite field analogues of certain matrices related to the Gamma function. For example, let $q=p^n$ be an odd prime power with $p$ prime and $n\in\mathbb{Z}^+$. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $\chi$ be a generator of the group of all mutiplicative characters of the finite field $\mathbb{F}_q$. For the Gauss sum $$G_q(\chi^{r})=\sum_{x\in\mathbb{F}_q}\chi^{r}(x)\zeta_p^{{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(x)},$$ we prove that $$\det \left[G_q(\chi^{2i+2j})\right]_{0\le i,j\le (q-3)/2}=(-1)^{\alpha_p}\left(\frac{q-1}{2}\right)^{\frac{q-1}{2}}2^{\frac{p^{n-1}-1}{2}},$$ where $$\alpha_p= \begin{cases} 1 & \mbox{if}\ n\equiv 1\pmod 2, (p^2+7)/8 & \mbox{if}\ n\equiv 0\pmod 2. \end{cases}$$