I know the following theorem:
Let $f(x)\in\mathbb{Z}[x]$ be a monic separable polynomial of degree $n$. If $f(x)\mod p$ is separable and irreductible for some prime $p$, then there is a cycle of lenght $n$ in $\text{Gal}(f,\mathbb{Q})\subset S_n$.
However, it seems that more than that is true. Lets say we reduce $f(x)$ modulo $p$ and the result is equal to $$f_1(x)f_2(x)\dotsc f_m(x),$$ where each $f_i(x)$ is irreductible, separable and of degree $n_i$. Then I think there exists cycles of lenght $n_i$ for all $i$ in $S_n$.
Is that true? If so, why?