apologies if this is a naive question. Consider two Galois extensions, K and L, of the rational numbers. For each extension, consider the set of rational primes that split completely in the extensions, say Split(K) and Split(L).

If Split(K) = Split(L), then is it necessarily true that K and L are isomorphic as Galois extensions of the rationals?

If so, for a given set of rational primes, S, is there a way to construct the extension over which S is the set of completely split primes?

References welcomed! Thanks, Martin