Let $f$ be an irreducible monic polynomial over $\Bbb Z$ and $L$ its splitting field. Let $p$ be a prime number and $\frak p$ be a prime ideal of ${\cal O} _L$ lying over $p$ (the question also makes sense in a more general setting, but lets consider this case for simplicity).
My question is more or less:
Is ${\cal O} _L/\frak p$ the splitting field of $\bar{f}$, the reduction of $f$ modulo $p$?
After thinking about this a bit I figured out that this is not true in general. Consider e.g. $f=X^2-5, p=2$.
But I think the statement is true if we impose the additional condition that $f$ is separable modulo $p$, i.e. $p$ doesn't divide the discriminant $D(f)$ of $f$.
Let $R=\Bbb Z[\alpha_1,\dots,\alpha_n]$ where the $\alpha_i$ are the roots of $f$. Then clearly $R/R\cap\frak p$ is the splitting field of $\bar{f}$ over $\Bbb F_p$. Our claim would follow if we could prove that $p$ does not divide the index $[{\cal O}_L : R]$, because in this case the map $R/R\cap{\frak p} \to {\cal O}_L / \frak p$ would be an isomorphism.
I tried out some examples and every time the discriminant of $f$ had the same prime factors as the discriminant of $R$, so the above seems to be true.
Maybe there is some formula for the discriminant of the product of orders? In this case we could use the fact that the discriminant of $\Bbb Z[\alpha_i]$ is (up to sign) the discriminant of $f$.
Note: I also saw this question but it didn't really answer my question, I think the answer there refers to something linked in the comments which is no longer accessible.
There is a $p$-adic way, possibly more enlightening than the relative discriminant and the index $[O_L:R]$:
The completion $L_\mathfrak{p}$ is the splitting field of $f\in \Bbb{Q}_p[x]$ and $(O_L)_\mathfrak{p}/\mathfrak{p}\cong O_L/\mathfrak{p}$.
When $f\in \Bbb{Z}_p[x]_{monic}$ is separable $\bmod p$ then Hensel lemma gives that its splitting field is $\Bbb{Q}_p(\zeta_n)$ where $\Bbb{F}_p(\zeta_n)$ is the splitting field of $f\bmod p$.
Whence $$O_L/\mathfrak{p}\cong O_{\Bbb{Q}_p(\zeta_n)}/(p)\cong \Bbb{F}_p(\zeta_n)$$