Let $K$ be a number field, and $K’/K$ a finite, unramified Galois extension. Let $L/K$ be an arbitrary finite Galois extension. Is it then true that $LK’/L$ is unramified? Here $LK’$ is the compositum of $L$ and $K,$ taken inside of some algebraic closure.
This is clearly true for local fields, and I have an argument for why we should be able to reduce to that situation, but I still feel a bit uneasy so I am just after knowing whether this is true. It is of course OK to post a proof of why this is so, but I am mostly after either a confirmation that this is the case, or a counterexample.