ON CYCLOTOMIC MATRICES INVOLVING GAUSS SUMS OVER FINITE FIELDS

Inspired by the works of L. Carlitz [Acta Arith. 5 (1959), pp. 293- 308] and Z.-W. Sun [Finite Fields Appl. 56 (2019), pp. 285-307] 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 = pn be an odd prime power with p prime and n is an element of Z+. Let zeta p = e2 pi i/p and let chi be a generator of the group of all multiplicative characters of the finite field Fq. For the Gauss sum we prove that where E Gq(chi r) = x is an element of Fq chi r (x)zeta TrFq/Fp (x) p , det [Gq(chi 2i+2j))0 <= i,j <=(q-3)/2 = (-1)alpha alpha n = (q -1)q-1 2 pn-1-1 n 2 2 , 2 Gamma 1 if n equivalent to 1 (mod 2), (p2 + 7)/8 if