Let $\Omega$ be a matrix with entries $a_{jk}=\omega^{jk}$, where $0\leq j,k\leq n-1$, and $\omega=e^{-2\pi i/N}$, with $N\in \mathbb{N}$, so $\Omega$ looks like $$ \Omega=\begin{pmatrix} 1 & 1 & \cdots & 1 \\ 1 & \omega & \cdots & \omega^{n-1}\\ 1 & \omega^2 & \cdots & \omega^{2(n-1)}\\ \vdots & \vdots & \vdots & \vdots \\ 1 & \omega^{n-1} &\cdots & \omega^{(n-1)(n-1)} \end{pmatrix} $$
Suppose that we take a minor $A$ of $\Omega$, that is, $A=(w^{jk})$, with $j=a_1,a_2,\ldots , a_c$, and $k=b_1,b_2,\ldots , b_d$, with $a_j$ and $b_j$ pairwise different and obviously $0\leq a_j,b_k\leq n-1$, for all $j,k$.
My question is: what can we say about rank $A$? There is a well-known theorem due to Chebotarev which states that $A$ has always maximum rank whenever $N$ is prime (actually, its formulation is that every minor of $\Omega$ is non-zero). I also found a helpful paper which works what I am asking for (http://www.math.bgu.ac.il/~pakovich/Publications/rem.pdf), but in a different way. It states the following:
''For a complex polynomial $P(z)$ denote by $w(P)$ the number of non-zero coefficients of $P(z)$. It is easy to see that Chebotarev theorem is equivalent to the following statement: if a non-zero polynomial $P(z)$, $\deg P(z)\leq N-1$, has $k$ different roots which are roots of unity then $w(P)>k$ whenever $N$ is prime.''
I have not been able to prove this equivalence, and therefore, I cannot interpret the theorem give in this paper in terms of the ranks of the submatrices of $\Omega$ (whenever $N$ is not prime). Can anyone give me some hint about how to prove this equivalence or the interpretation of it when related to matrices? Thanks in advance.