I am mainly using these notes on algebraic number theory. Let $L/K$ be a finite degree field extension given by $L=K(\alpha)$ (this is not restrictive by the primitive element theorem). Let $\mathfrak{p}$ be a prime of $\mathcal{O}_K$. Further let $f$ be the minimal polynomial of $\alpha$ and $f=\prod_j g_j^{e_j}$ for irreducible polynomials $g_j$ over $\mathcal{O}_K/\mathfrak{p}$.
If $\mathcal{O}_L=\mathcal{O}_K[\alpha]$, then we can apply the Dedekind-Kummer Theorem that for all primes $\mathfrak{p}\subset K$,
$\mathfrak{p}\mathcal{O}_L=\prod_{j} \mathfrak{q_j}^{e_j}$, where $\mathfrak{q_j}=(\mathfrak{p},g_j(\alpha))$.
Further, from remark 6.34, we have that in the case of $K=\mathbb{Q}$, that this theorem holds more broadly (with $p$ a rational prime) for all $(p)\mathcal{O}_L$ as long as $p \nmid [\mathcal{O}_L:\mathbb{Z}[\alpha]]$.
Question: What is a nice example of an algebraic number field $L=(\mathbb{Q}[x]/(f))/\mathbb{Q}=\mathbb{Q}[\alpha]/\mathbb{Q}$ such that $\mathcal{O}_L\neq \mathbb{Z}[\alpha]$, and a prime $(p)\subset \mathbb{Q}$ such that $p | [\mathcal{O}_L:\mathbb{Z}[\alpha]]$ and furthermore such that one CANNOT apply Dedekind-Kummer to factor. More specifically, what does the factorization of $(p)\mathcal{O}_L$ look like?