Let $K$ be a number field, with ring of integers $O_K$, and let $\alpha\in O_K$ be a primitive element for the extension $K/Q$, with minimal monic polynomial $f(x)\in Z[x]$. If $p$ is a prime number, then the factorisation of the principal ideal $(p)$ in the Dedekind ring $O_K$ can be described in terms of the reduction mod $p$ of $f(x)$ as long as $p$ does not divide the (finite) index $[O_K:Z[\alpha]]$. This is well known and rather elementary, as one can say.

On the other hand, I have never seen in the literature a more general statement relating the factorisation of $(p)$ with the reduction mod $p$ of $f(x)$ without the assumption that $p$ did not divide $[O_K:Z[\alpha]]$. In this more general setting, the sole mod $p$ reduction of $f(x)$ does not give enough information about the splitting of $(p)$, one does need some extra "characteristic zero" information. Would the $p$-adic valuation of the discriminant of $f(x)$ be enough? At least if one knows that $p$ is tamely ramified in $K$. Have you ever seen this discussed or have you ever thought about it? Many thanks.