On the cyclotomic field $\mathbb{Q}(e^{2π{\bf i}/p})$ and Zhi-Wei Sun's conjecture on $\det M_p$

TL;DR

This work resolves Zhi-Wei Sun's conjecture on the determinant $\det M_p$ of a Legendre-symbol matrix by embedding the problem in the $p$-th cyclotomic field. The authors first prove that $\det M_p$ is a unit in $\mathbb{Z}[\zeta]$ by employing Vsemirnov's decomposition and expressing $\det M_p$ via products and adjugates of carefully constructed matrices, reducing the core issue to showing a certain auxiliary determinant $\det H$ is a unit. They then determine $\det M_p$ modulo $p$ by transferring to the prime ideal $\mathfrak{p}$ and leveraging Gauss sums, classical products, and Sun's results on quadratic-residue-related products. The final explicit values are $\det M_p = (-1)^{\frac{p-1}{4}}$ for $p\equiv 1\pmod{4}$ and $\det M_p = (-1)^{\frac{h(-p)-1}{2}}$ for $p\equiv 3\pmod{4}$, with $h(-p)$ the class number of $\mathbb{Q}(\sqrt{-p})$. This ties determinants of Legendre-symbol matrices to cyclotomic units and the arithmetic of imaginary quadratic fields, providing a complete resolution of Sun's conjecture and enriching connections between Gauss sums, class numbers, and determinant theory.

Abstract

In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of $p$-th cyclotomic field and the finite field $\mathbb{F}_p$, we confirm this conjecture.

Theorems & Definitions (4)

• Conjecture 1.1
• Theorem 1.1
• Remark 1.1
• Lemma 2.1