On cyclotomic matrices involving Gauss sums over finite fields

Inspired by the works of L. Carlitz and Z.-W. Sun 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 $q=p^n$ be an odd prime power with $p$ prime and $n\in\mathbb{Z}^+$. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $\chi$ be a generator of the group of all mutiplicative characters of the finite field $\mathbb{F}_q$. For the Gauss sum $$G_q(\chi^{r})=\sum_{x\in\mathbb{F}_q}\chi^{r}(x)\zeta_p^{{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(x)},$$ we prove that $$\det \left[G_q(\chi^{2i+2j})\right]_{0\le i,j\le (q-3)/2}=(-1)^{\alpha_p}\left(\frac{q-1}{2}\right)^{\frac{q-1}{2}}2^{\frac{p^{n-1}-1}{2}},$$ where $$\alpha_p= \begin{cases} 1 & \mbox{if}\ n\equiv 1\pmod 2, (p^2+7)/8 & \mbox{if}\ n\equiv 0\pmod 2. \end{cases}$$