Let $n>1$ be a natural number.
Question. Let $K$ be a number field, and let $S\subset V_f(K)$ be a finite set of finite (non-archimedean) places of $K$. Does there exist a normal extension $L/K$ of degree $n$ such that for all places $v\in S$, the tensor product $L\otimes_K K_v$ is a field (a normal extension of $K_v$ of degree $n$)? Here $K_v$ denotes the completion of $K$ at $v$.