On certain cyclotomic matrices involving biquadratic residues

Abstract

Let $p\equiv 1\ ({\mathrm{mod}}\ 4)$ be a prime and let ${\mathfrak {p}}$ be a prime ideal of ${\mathbb {Z}}[\sqrt{-1}]$ with $p\in {\mathfrak {p}}$. Let $(\frac{\cdot }{{\mathfrak {p}}})_4$ be the biquadratic residue symbol modulo ${\mathfrak {p}}$. Let $0<a_1<a_2<\cdots<a_{(p-1)/4}<p$ be all the biquadratic residues modulo p in the interval (0, p). In this paper, we study some arithmetic properties of the cyclotomic matrix

$$\begin{aligned} W_{{\mathfrak {p}}}=\bigg [\left( \frac{a_i+a_j}{{\mathfrak {p}}}\right) _4\bigg ]_{1\le i,j\le (p-1)/4}. \end{aligned}$$