Counting matrices with a fixed primary polynomial
= Counting matrices with a fixed primary polynomial
Fix an irreducible polynomial f of degree d with nonzero constant term. In $GL_{dm}(q)$, the number of matrices whose minimal polynomial is a power of f is $|GL_{dm}(q)|q^{d(m^2-m)}/|GL_m(q^d)|$. 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>.