Let $L/K$ be an extension of number fields. Take a prime $\mathfrak{p} \in \text{Spec}(\mathcal{O}_K)$. I suspect that $\mathcal{O}_L$ is unramified over $\mathcal{O}_K$ at $\mathfrak{p}$ if and only if $\mathcal{O}_{L_{\mathfrak{p}}}$ is étale over $\mathcal{O}_{K_{\mathfrak{p}}}$, where $L_{\mathfrak{q}}$ is the completion of $L$ at a prime $\mathfrak{q}$ occuring in the prime factorization of $\mathfrak{p} \mathcal{O}_L = \mathfrak{q}_1^{e_1} \cdots \mathfrak{q}_n^{e_n}$.
Can anyone show this?
This could be interesting since It would help to understand the picture of the étale fundamental group. $\text{Gal}_{ab}(\overline{\mathbb{Q}}/\mathbb{Q})$ has $p$-unramified quotient group for each prime. I suspect that, for each prime $p$, this group corresponds to the absolute galois groups of $\mathbb{Q}_p$, i.e. $\widehat{\mathbb{Q}_p^{\times}}$. Since the absolute galois group of $\mathbb{Q}$ is the finite adeles, this is talking about the quotient map $\prod'_{p} \mathbb{Q}_p \rightarrow \mathbb{Q}_p$, where $'$ denotes a subproduct.