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 $q\equiv 3(\mathrm{mod}4)$ be an odd prime power and let $\varphi$ 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 $$\det {\left[t+\varphi (s_i+s_j)+\varphi (s_i-s_j)\right]}_{1\le i,j\le (q-1)/2}=\left(\frac{q-1}{2}t-1\right)q^{\frac{q-3}{4}}.$$ This confirms a conjecture of Zhi-Wei Sun.

References in corpus (1)

• Problems and results on determinants involving Legendre symbols

Cited by in corpus (1)

• Gaussian hypergeometric functions and cyclotomic matrices