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arXiv · 2405.08552

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

Jie Li·Hai-Liang Wu
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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.

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Jie Li, Hai-Liang Wu. 2024-05-14. A conjecture of Zhi-Wei Sun on matrices concerning multiplicative subgroups of finite fields. https://doi.org/10.1017/s0004972724000765

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