Total ramification from an Eisenstein polynomial
= Total ramification from an Eisenstein polynomial
If an algebraic integer $\alpha$ has a degree-$n$ minimal polynomial that is Eisenstein at $p$, then $p$ has a unique prime ideal $P$ above it in $\mathbb Q(\alpha)$, with
$$
(p)=P^n,\qquad v_P(\alpha)=1.
$$
This conclusion concerns the full <ring of integers of a number field> and does not assume that it equals $\mathbb Z[\alpha]$.