For N=p^m, det of the cyclotomic Gauss-sum matrix A_k(χ) equals (±) d^d times the constant term of the minpoly of the Gauss period η_N(k), and vanishes iff p divides k.
Newman, Roots of unity and covering sets, Math
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The Gauss periods and cyclotomic matrices involving Gauss sums over cyclic groups
For N=p^m, det of the cyclotomic Gauss-sum matrix A_k(χ) equals (±) d^d times the constant term of the minpoly of the Gauss period η_N(k), and vanishes iff p divides k.