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 $1=s_1,s_2,\cdots ,s_{(q-1)\mathrm{/}2}$ be all the nonzero squares over ${\mathbb{F}}_q$, where $q=p^f$ is an odd prime power with $q\ge 7$. We prove that the matrix $$B_q((q-3)\mathrm{/}2)={\left[{(s_i+s_j)}^{(q-3)\mathrm{/}2}\right]}_{2\le i,j\le (q-1)\mathrm{/}2}$$ is a singular matrix whenever $f\ge 2$. Also, for the case $q=p$, we show that $$\det B_p((p-3)\mathrm{/}2)=0\iff Q_p\equiv 2(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2\mathbb{Z}),$$ where $Q_p$ is the $p$-th term of the companion Pell sequence $\{Q_i{\}}_{i=0}^{\mathrm{\infty }}$ defined by $Q_0=Q_1=2$ and $Q_{i+1}=2Q_i+Q_{i-1}$.