= Prime Fourier matrix minor theorem
{title2=$\det[\zeta^{r_ic_j}]\ne0$}
For a prime $p$, distinct row and column residues and $\zeta=e^{2\pi i/p}$, every square <discrete Fourier transform> submatrix is nonsingular. Sort the selected column exponents decreasingly as $b_i$ and write $b_i=\lambda_i+N-i$. Its <determinant> factors by the <bialternant formula> into a nonzero <Vandermonde determinant> and $s_\lambda(\zeta^{r_1},\ldots,\zeta^{r_N})$. Modulo $1-\zeta$, this Schur factor is $s_\lambda(1^N)$ in $\mathbb F_p$. The <Schur evaluation at all ones> product has numerator and denominator differences strictly between zero and $p$, so its residue is nonzero. The Schur factor, and hence the minor, cannot vanish.
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