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

In this paper, we study certain determinants over finite fields. Let F q be the finite field of q elements with q an odd prime power and q ≡ 2 ( mod 3 ) . Let a 1 , a 2 , ⋯ , a q - 1 be all nonzero elements of F q and let T q = 1 a i 2 - a i a j + a j 2 1 ≤ i , j ≤ q - 1 be a matrix over 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.