A conjecture of Zhi-Wei Sun on determinants over finite fields

TL;DR

This paper determines the explicit value of a specific determinant over finite fields and confirms a conjecture by Zhi-Wei Sun related to these determinants.

Contribution

The paper provides an explicit calculation of the determinant of a matrix over finite fields and proves a conjecture proposed by Zhi-Wei Sun.

Findings

Explicit value of the determinant over finite fields is obtained.

Confirms Zhi-Wei Sun's conjecture on determinants.

Provides new insights into determinants over finite fields.

Abstract

In this paper, we study certain determinants over finite fields. Let $\mathbb{F}_q$ be the finite field of $q$ elements and let $a_1,a_2,\cdots,a_{q-1}$ be all nonzero elements of $\mathbb{F}_q$. Let $T_q=\left[\frac{1}{a_i^2-a_ia_j+a_j^2}\right]_{1\le i,j\le q-1}$ be a matrix over $\mathbb{F}_q$. We obtain the explicit value of $\det T_q$. Also, as a consequence of our result, we confirm a conjecture posed by Zhi-Wei Sun.