 Counting matrices with a fixed primary polynomial

Fix an irreducible polynomial f of degree d with nonzero constant term. In $G{L}_{dm}(q)$, the number of matrices whose minimal polynomial is a power of f is $|G{L}_{dm}(q)|q^{d(m^2-m)}/|G{L}_m(q^d)|$. The primary matrix centralizer formula is identical to the unipotent centralizer formula over $F_{q^d}$, and summing reciprocal centralizer orders reduces to counting nilpotent matrices by the Fitting decomposition.