This is a fun exercise I came with.
Let $K$ be a number field, $O_K$ non-monogenic and $\mathfrak{p}$ a prime ideal above a bad prime $p$. Then how would you show that for some $a\in O_K,\pi\in \Bbb{Z}[a]$ $$O_{K,\mathfrak{p}} = \Bbb{Z}[a]_{(p,\pi)}\ ?$$
Here bad prime means that $O_K/(p)$ is not isomorphic to $\Bbb{F}_p[x]/(f(x))$, for example $p=2$ and $K=\Bbb{Q}[T]/(T^3-T^2-2T-8)$.