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

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.