Chebotarev's theorem on roots of unity says that all the minors of a prime-length DFT matrix over the complex numbers are nonzero. I was wondering if there was an analogue for finite fields.
More precisely, let $p$ be prime and $\omega=e^{2\pi i/p}$, the complex $p$th root of unity, and let $\Omega$ be the matrix given by $\Omega_{ij}=\omega^{ij}$, for $i,j \in \{0,1,\dots,p-1\}$. Then Chebotarev's theorem on roots of unity says that every square submatrix of $\Omega$ is nonsingular; see wikipedia.
Alternatively, this is equivalent to stating that, for a complex-valued polynomial $f$ of degree less than $p$, if $f$ evaluates to zero for $t$ distinct powers of $\omega$, then either $f$ is zero, or $f$ has at least $t+1$ terms, or in other words, $$|\mathrm{supp}(f)| + |\mathrm{supp}(\hat{f})| > p.$$
Suppose then that $q$ is a prime power for which $p$ divides $q-1$, such that $\mathbf{F}_q$ contains a primitive $p$th root of unity $\omega$. Does the analogue hold there? I am particularly interested in the case that $q$ is prime.
The $2 \times 2$ square submatrices will be nonsingular because a polynomial $x^a + cx^b$, where $p>a>b\geq 0$, $c \neq 0$ will have a root $\omega^i$ if and only if $\omega^{(a-b)i}=c$, which will hold for at most one $i \in [p]$. Computation shows the analogue holds for the $p=7$th roots of unity modulo $q=29$.