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

Jie Li, Hai-Liang Wu

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 $φ$ 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 $$\det\left[t+φ(s_i+s_j)+φ(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}=\left(\frac{q-1}{2}t-1\right)q^{\frac{q-3}{4}}.$$ This confirms a conjecture of Zhi-Wei Sun.

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 be an odd prime power and let be the unique quadratic multiplicative character of the finite field . If set , then we prove that This confirms a conjecture of Zhi-Wei Sun.
Paper Structure (2 sections, 3 theorems, 39 equations)

This paper contains 2 sections, 3 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Thm. Aq(k)']}

Key Result

Theorem 1.1

Let $q=p^s$ be an odd prime power with $p$ prime and $s\in\mathbb{Z}^+$. Then for any positive integer $k\mid q-1$, the following results hold. (i) Suppose $q\equiv 1\ ({\rm{mod}}\ 2k)$. Then $\det A_k(t)=0$. In particular, in the case $q\equiv1\ ({\rm{mod}}\ 4)$, we have $\det A_2(t)=0$. (ii) If $q (iii) Suppose $q\equiv 1\ ({\rm{mod}}\ 4)$ and $q\not\equiv1\ ({\rm{mod}}\ 2k)$. Then there is an i

Theorems & Definitions (5)

  • Conjecture 1.1: Sun
  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2