Is there a finite set of primes, $S \subset Spec(\mathbb{Z})$ such that if $K$ over $\mathbb{Q}$ is a number field such that every $p \in S$ is completely split in $K$ then $K=\mathbb{Q}$? If so, what do we know about such $S$'s?



Is there a finite set of primes such that if K over Q is completely split at all those primes then it is Q?
This is not even true for quadratic extensions. Given primes $p\_1, p\_2, ... p\_n$ find a prime $q \equiv 1 \bmod 4p\_1 ... p\_n$, which exists by Dirichlet's theorem, and consider $K = \mathbb{Q}(\sqrt{q})$.


