Principal minors of Fourier matrices of square-free order

TL;DR

This paper investigates when all principal minors of Fourier matrices of square-free order are non-zero, extending Chebotarev's theorem beyond prime orders and providing new insights into their structure.

Contribution

It establishes conditions under which all principal minors of Fourier matrices of square-free order are non-zero, advancing understanding of their algebraic properties.

Findings

All principal minors are non-zero for certain square-free orders

Extends Chebotarev's theorem to non-prime square-free orders

Lays groundwork for analyzing composite orders with squares in future work

Abstract

Chebotarev's theorem on roots of unity states that all minors of a Fourier matrix are non-zero if and only if the order of the matrix is prime. We establish cases in which all principal minors of Fourier matrices of square-free order are non-zero. In a subsequent paper we discuss the case of composites containing squares.