Real and finite field versions of Chebotarev's theorem

TL;DR

This paper generalizes Chebotarev's theorem to real and finite fields, removing previous restrictions and broadening its applicability to problems like phase retrieval and Riesz bases of exponentials.

Contribution

It extends Chebotarev's theorem to real and finite fields, eliminating order restrictions and lowering characteristic bounds, with implications for phase retrieval and exponential bases.

Findings

Generalized Chebotarev's theorem to real fields.

Extended finite field version without order restrictions.

Lowered characteristic bounds for finite field case.

Abstract

Chebotarev's theorem on roots of unity states that all minors of the Fourier matrix of prime size are non-vanishing. This result has been rediscovered several times and proved via different techniques. We follow the proof of Evans and Isaacs and generalize the original result to a real version and a version over finite fields. For the latter, we are able to remove an order condition between the characteristic of the field and the size of the matrix as well as decrease a sufficient lower bound on the characteristic by Zhang considerably. Direct applications include a specific real phase retrieval problem as well as a recent result for Riesz bases of exponentials.