A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields

TL;DR

This paper proves a conjecture by Zhi-Wei Sun regarding the determinant of a matrix constructed from quadratic residues and the quadratic character over finite fields, revealing a precise formula for the determinant.

Contribution

It confirms Zhi-Wei Sun's conjecture by deriving an explicit formula for the determinant of a matrix involving quadratic residues and characters over finite fields.

Findings

Determinant formula involving quadratic residues and characters

Confirmation of Sun's conjecture for specific finite fields

Explicit expression for the matrix determinant

Abstract

Motivated by the recent work of Zhi-Wei Sun on determinants involving the Legendre symbol, in this paper, we study some matrices concerning subgroups of finite fields. For example, let $q\equiv 3\pmod 4$ be an odd prime power and let $\phi$ be the unique quadratic multiplicative character of the finite field $\mathbb{F}_q$. If set $\{s_1,\cdots,s_{(q-1)/2}\}=\{x^2:\ x\in\mathbb{F}_q\setminus\{0\}\}$, then we prove that $$\det\left[t+\phi(s_i+s_j)+\phi(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}=\left(\frac{q-1}{2}t-1\right)q^{\frac{q-3}{4}}.$$ This confirms a conjecture of Zhi-Wei Sun.