Suppose that I'm given a set of rational primes $S$ with positive Dirichlet density, and a finite set of primes $R$, disjoint from $S$. 
Does there exist a number field $K$ that is;


* unramified outside $R$;
* splits only at primes in some subset $S'\subseteq S$?


In my situation, I have a semisimple representation $\rho$ of $G\_\mathbb Q =\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$, unramified outside $R$, and I have information about the action of Frobenius elements $\mathrm{Frob}\_p$ for $p\in S$ (let's say I know their characteristic polynomials). I'd like to know if this completely determines my representation on some open subgroup of $G\_\mathbb Q$.



Extension of $\mathbb Q$ which splits only at primes in $S$
For many choices of $R$ and $S$ the answer is obviously no. For example, if $R$ is empty, then the answer is no, because there are no unramified extensions of $\mathbb{Q}$.


For a more interesting example, let $S$ be the set of primes $p \equiv 3 \pmod{4}$ and $R$ be any finite subset of $\{ 2, 5, 13, 17, 29, \ldots \}$. Then there is no number field that splits only at primes $S' \subseteq S$. This is because if $K/\mathbb{Q}$ is such a number field, any prime that splits in $K[i]$ must be $\equiv 1 \pmod{4}$, and $S$ doesn't contain any such primes.


