Let $K/\mathbb{Q}$ be a number field. We say that a rational prime $p$ splits in $K$ if there exists a prime $\mathfrak{p}$ of $K$ above $p$ of interia degree $1$.

Is a number field $K$ uniquely determined by the set of primes which split in $K$?

A well-known application of the Chebotarev density theorem (Neukirch Cor. 13.10) says that this is true when $K/\mathbb{Q}$ is *Galois* (note that here a prime splits if and only if it splits completely). So I am really interested in what happens in the non-Galois case.

Note also that the answer to the analogous question for completely split primes is *no*; indeed a prime is completely split in $K$ if and only if it is completely split in the Galois closure of $K$.