Let \(q=p^n\) be an odd prime power and let \(\mathbb {F}_q\) be the finite field with q elements. Let \(\widehat{\mathbb {F}_q^{\times }}\) be the group of all multiplicative characters of \(\mathbb {F}_q\) and let \(\chi \) be a generator of \(\widehat{\mathbb {F}_q^{\times }}\) . In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over \(\mathbb {F}_q\) . For example, let \(s_1,s_2,\ldots ,s_{(q-1)/2}\) be all nonzero squares over \(\mathbb {F}_q\) . For any integer \(1\le r\le q-2\) , define the matrix \(\begin{aligned} B_{q,2}(\chi ^r):=\left[ \chi ^r(s_i+s_j)+\chi ^r(s_i-s_j)\right] _{1\le i,j\le (q-1)/2}. \end{aligned}\) We prove that if \(q\equiv 3\ (\mathrm{{mod}}\ 4)\) , then \(\begin{aligned} \det (B_{q,2}(\chi ^r))= & \prod _{0\le k\le (q-3)/2}J_q(\chi ^r,\chi ^{2k})\\ = & {\left\{ \begin{array}{ll} (-1)^{\frac{q-3}{4}} \textbf{i}^nG_q(\chi ^r)^{\frac{q-1}{2}}/\sqrt{q} & \quad \text{ if }\; r\equiv 1\ (\mathrm{{mod}}\ 2),\\ G_q(\chi ^r)^{\frac{q-1}{2}}/q & \quad \text{ if }\; r\equiv 0\ (\mathrm{{mod}}\ 2), \end{array}\right. } \end{aligned}\) where \(J_q(\chi ^r,\chi ^{2k})\) and \(G_q(\chi ^r)\) are the Jacobi sum and the Gauss sum over \(\mathbb {F}_q\) respectively.
