On generalized Legendre matrices involving roots of unity over finite fields

Abstract: In this paper, motivated by the work of Chapman, Vsemirnov and Sun et al., we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting $a_1,\cdots,a_{(q-1)/2}$ be all non-zero squares in the finite field $\mathbb{F}q$ which contains $q$ elements with $2\nmid q$, we give the explicit value of $D{(q-1)/2}=\det[(a_i+a_j)^{(q-3)/2}]_{1\le i,j\le (q-1)/2}$. In particular, if $q=p$ is a prime greater than $3$, then $$\left(\frac{\det D_{(p-1)/2}}{p}\right)= \begin{cases} 1 & \mbox{if}\ p\equiv1\pmod4, (-1)^{(h(-p)+1)/2} & \mbox{if}\ p\equiv 3\pmod4\ \text{and}\ p>3, \end{cases}$$ where $(\cdot/p)$ is the Legendre symbol and $h(-p)$ is the class number of $\mathbb{Q}(\sqrt{-p})$.