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For the fixed degree-two irreducible polynomial $f$, a matrix with minimal polynomial a power of $f$ has only one primary component. Its conjugacy classes correspond to integer partitions $\lambda$ of $m$. By the primary matrix centralizer formula, its centralizer has exactly the same order $c_{\lambda }(q^2)$ as the centralizer of the corresponding unipotent matrix in $G{L}_m(q^2)$.

Sum the conjugacy class sizes:The same sum times $|G{L}_m(q^2)|$ is the number of unipotent matrices of $G{L}_m(q^2)$, which 5(c) shows is $(q^2{)}^{m^2-m}$. Hence counting matrices with a fixed primary polynomial givesIn particular the exponent $a$ of the minimal polynomial is not fixed in advance: summing all integer partitions includes every possible largest block size. The usual statement takes $m\ge 1$; dimension zero can instead be included with its identity convention.