The Pell sequence and cyclotomic matrices involving squares over finite fields

In this paper, by some arithmetic properties of the Pell sequence and some $p$-adic tools, we study certain cyclotomic matrices involving squares over finite fields. For example, let $1=s_1,s_2,\cdots,s_{(q-1)/2}$ be all the nonzero squares over $\mathbb{F}_{q}$, where $q=p^f$ is an odd prime power with $q\ge7$. We prove that the matrix $$B_q((q-3)/2)=\left[\left(s_i+s_j\right)^{(q-3)/2}\right]_{2\le i,j\le (q-1)/2}$$ is a singular matrix whenever $f\ge2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)/2)=0\Leftrightarrow Q_p\equiv 2\pmod{p^2\mathbb{Z}},$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i\}_{i=0}^{\infty}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.