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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 $s_{\lambda }({\zeta }^{r_1},\dots ,{\zeta }^{r_N})$. 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.

Put $b_i={\lambda }_i+N-i$. In the bialternant formula, substitute $x_j=e^{t{y}_j}$ with distinct $y_j$ and compare the first nonzero determinant coefficients as $t\to 0$. They arise from the distinct degrees $0,\dots ,N-1$ in the exponential expansions. The common factor in the $y_j$ and the factorials cancel, leaving the product of differences of the $b_i$ divided by the differences of $N-i$. This proves the displayed integral specialization without dividing by a vanishing Vandermonde value.