Exercise 11, § 12

Let f be a monic polynomial in $\mathbf{Z}[X]$, irreducible in $\mathbf{Q}[X]$ and let p be a prime number such that the canonical image $\overline{f}$ of f in $\mathbf{F}_p[X]$ is a polynomial without multiple roots. If $\Gamma$ (resp. $\Gamma'$) is the Galois group of the polynomial f (resp. $\overline{f}$) over $\mathbf{Q}$ (resp. $\mathbf{F}_p$), define an isomorphism of a subgroup of $\Gamma$ onto $\Gamma'$. Deduce that if $\overline{f} = g_1 g_2 \ldots g_r$ where the $g_j$ are irreducible in $\mathbf{F}_p[X]$ and if $g_j$ is of degree $n_j$ (so that f has degree $n = n_1 + n_2 + \ldots + n_r$), then the group $\Gamma$, qua subgroup of $\mathfrak{S}_n$ contains a product $\sigma_1 \sigma_2 \ldots \sigma_r$ of cycles with disjoint supports, $\sigma_j$ being of order $n_j$ for $1 \leq j \leq r$ (cf. I, p. 62). Generalize to the case where f is not monic, but where its leading coefficient is not divisible by p.