A conjecture of Zhi-Wei Sun on 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 $T_q={\left[\frac{1}{a_i^2-a_i{a}_j+a_j^2}\right]}_{1\le i,j\le q-1}$ 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.