If $K$ is a number field, does there always exist an unramified extension of $K$, i.e. an extension $L/K$ so that the extension is unramified at all $\mathfrak{p} \subseteq \mathcal{O}_K$? For example, given $K=\mathbb{Q}(\sqrt{-2})$, we have $\mathcal{O}_K=\mathbb{Z}[\sqrt{-2}]$. How would one construct a unramified extension?
I know ramification indexes and inertia degrees are multiplicative and a prime ramifies in $K$ if and only if it divides the discriminant of $K$ but I do not see how this helps me.
EDIT: As LordShark helpfully pointed out, these do not always exist. What I am really interested in is how does one construct them assuming they exist. Looking up the narrow class group, I think it should exist in the case I have above. So let's just say $K=\mathbb{Q}(\sqrt{d})$ for some square free $d \in \mathbb{Z}$, i.e. $d=-2,-5,7$, etc. for which such an extension exists.