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.

翻译：切博塔廖夫关于单位根的理论指出，素数阶傅里叶矩阵的所有子式均非零。该结论已被多次重新发现，并藉由不同技术得到证明。我们遵循埃文斯和艾萨克斯的证明方法，将原始结果推广至实域版本与有限域版本。对于后者，我们能够消除域特征与矩阵阶数之间的阶条件，并大幅降低了张所提出的特征值充分下界。直接应用包括特定的实相位恢复问题，以及关于指数函数里兹基的最新结果。