A CONJECTURE OF ZHI-WEI SUN ON MATRICES CONCERNING MULTIPLICATIVE SUBGROUPS OF FINITE FIELDS

Abstract Motivated by the recent work of Zhi-Wei Sun [‘Problems and results on determinants involving Legendre symbols’, Preprint, arXiv:2405.03626], 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 the set $\{s_1,\ldots ,s_{(q-1)/2}\}=\{x^2:\ x\in \mathbb {F}_q\setminus \{0\}\}$ , then we prove that $$ \begin{align*}\det[t+\phi(s_i+s_j)+\phi(s_i-s_j)]_{1\le i,j\le (q-1)/2}=\bigg(\frac{q-1}{2}t-1\bigg)q^{{(q-3)}/{4}}.\end{align*} $$ This confirms a conjecture of Zhi-Wei Sun.