= Solution
Let $L=\mathbb Q(\alpha)$, where $\alpha$ is an <algebraic integer> with monic <minimal polynomial> $f\in\mathbb Z[X]$, and put $I=[\mathcal O_L:\mathbb Z[\alpha]]$. The <Kummer-Dedekind theorem> applies to a <prime number> $p$ with $p\nmid I$. Write its reduction as a product of <polynomial factors> over $\mathbb F_p$:
$$
\overline f=\prod_{j=1}^r g_j^{e_j},
$$
where the $g_j$ are distinct monic <irreducible polynomials>, choose monic integral lifts $G_j$. Then the distinct <prime ideals> above $p$ are $\mathfrak p_j=(p,G_j(\alpha))$, and
$$
\boxed{p\mathcal O_L=\prod_{j=1}^r\mathfrak p_j^{e_j},\qquad f(\mathfrak p_j/p)=\deg g_j.}
$$
Thus the multiplicity of a factor is its <ramification index of a prime ideal>, and its degree is the <residue degree>. The hypothesis on the <order in a number field> is essential: one cannot infer <prime ideal factorization> from an arbitrary defining polynomial at a prime dividing its index. There is also a relative version over a base <number field> $K$: if $\mathcal O_L$ and $\mathcal O_K[\alpha]$ agree after localization at $\mathfrak p$, factor the <minimal polynomial> over $\mathcal O_K/\mathfrak p$ and use $(\mathfrak p,G_j(\alpha))$ with the same conclusions.
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