Distribution of decomposition types of primes in non-Galois extensions of number fields

Question

Let $L/K$ be an extension of number fields. If $p$ is a prime of $K$ that is unramified in $L/K$ and $(f_1, \dots, f_r)$ is a partition of $n = [L:K]$, say that $p$ has "decomposition type" $(f_1, \dots, f_r)$ if there are $r$ primes of $L$ lying over $p$, with inertia degrees $f_1, \dots, f_r$.

Is it true that the set of primes of a fixed decomposition type is either infinite or empty? For Galois extensions $L/K$ this is immediate from the Chebotarev density theorem, but what happens in general?

Answer 1

Let $F$ be the Galois closure of $L/K$, so $p$ is unramified in $F$. Let $H=Gal(F/L) \subset G=Gal(F/K), P$ a prime of $F$ over $p$, and $\phi=Frob(P/p) \in G$. Then $\{f_1,...,f_r\}$ are the sizes of the orbits of $\phi$ acting by translation on the cosets of $H$ in $G$, and the Chebotarev density theorem says that the set of primes $q$ of $K$ for which $Frob(Q/q)=\phi$ for some prime $Q/q$ of $F$ has density $=$ #conjugates of $\phi$ in $G/|G|$, and all such $q$ have the same "decomposition type" $\{f_1,...,f_r\}$.