Let $K$ be a number field and $S$ be a finite set of primes. Is it possible to construct a finite extension $M$ of $K$ such that $LM/M$ is unramified at (the primes above) $S$ for all degree $n$ extensions $L$ of $K$?
I think I can create an $M$ that works for all $L$ that are tamely ramified. Abhyankar's lemma says that (for local fields) if $L/K$ is tamely ramified and $e(L/K)$ divides $e(M/K)$, then $LM/M$ is unramified. So we can just take $M$ as the composition of totally ramified extensions at each prime in $S$ of degree $n!$, so that $e(L/K)$ always divide $e(M/K)=n!$. But I don't know if there is another approach that works in general.