A remark on the Chebotarev theorem about roots of unity

TL;DR

This paper explores an extension of Chebotarev's theorem, which originally states that all minors of a specific matrix are non-zero for prime n, to the case where n is composite, providing new insights into the structure of such matrices.

Contribution

The paper offers a novel analogue of Chebotarev's theorem applicable to composite integers n, expanding understanding beyond the prime case.

Findings

Established conditions under which minors of the matrix are non-zero for composite n

Extended Chebotarev's theorem to composite n cases

Provided theoretical framework for analyzing roots of unity matrices

Abstract

Let $\Omega$ be a matrix with entries $a_{i,j}=\omega^{ij},$ $1\leq i,j \leq n,$ where $\omega=e^{2\pi \sqrt{-1}/n},$ $n\in \mathbb N.$ The Chebotarev theorem states that if $n$ is a prime then any minor of $\Omega$ is non-zero. In this note we provide an analogue of this statement for composite $n.$