Real and finite field versions of Chebotarev's theorem
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.
Keywords
Cite
@article{arxiv.2506.02947,
title = {Real and finite field versions of Chebotarev's theorem},
author = {Tarek Emmrich and Stefan Kunis},
journal= {arXiv preprint arXiv:2506.02947},
year = {2025}
}
Comments
We sincerely thank Romanos Malikiosis for poining out a mistake in the previous version of Lemma 12 which had resulted in a flawed version of Corollary 17 for the nonprimitive case. The corrected version now is stated for the primitive case only and Section 4.1 partially extends our approach to the nonprimitive case