A remark on the Chebotarev theorem about roots of unity

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 $n.$