A Conjecture of Zhi-Wei Sun on Determinants Over Finite Fields

Abstract

In this paper, we study certain determinants over finite fields. Let ${\mathbb {F}}_\text {q}$ be the finite field of q elements with q an odd prime power and $q\equiv 2\pmod 3$. Let $a_1,a_2,\cdots ,a_{\text {q}-1}$ be all nonzero elements of ${\mathbb {F}}_\text {q}$ and let $T_\text {q}=\left[ \frac{1}{a_i^2-a_ia_j+a_j^2}\right] _{1\le i,j\le q-1}$ be a matrix over ${\mathbb {F}}_\text {q}$. We obtain the explicit value of $\det (T_\text {q})$. Also, as a consequence of our result, we confirm a conjecture posed by Zhi-Wei Sun.