Let $\alpha$ be a root of $f(X)=X^3-X+1$, $K=\mathbb{Q}(\alpha)$ and $L$ the splitting field of $f(X)$, so $Gal(L/\mathbb{Q})=S_3$. This is an old oral exam question and I'm trying to figure out how to determine the densities of different splittings in $K$.

We have the following options for primes $p\in\mathbb{Z}$ that are unramified in $L$:

- $(p)=\mathfrak{p}$ in $K\Rightarrow (p)=\mathfrak{P}_1\mathfrak{P}_2$ in $L$.
- $(p)=\mathfrak{p}_1\mathfrak{p}_2$ in $K\Rightarrow (p)=\mathfrak{P}_1\mathfrak{P}_2\mathfrak{P}_3$ in $L$.
- $(p)=\mathfrak{p}_1\mathfrak{p}_2\mathfrak{p}_3$ in $K\Rightarrow (p)=\mathfrak{P}_1\mathfrak{P}_2\mathfrak{P}_3$ or $(p)$ is totally split in $L$

The problem here is that (2) and (3) share a common type of splitting. Given a prime $\mathfrak{P}$ in $L$, $(p)=\mathfrak{P}\cap\mathbb{Z}$ and such that $(p)=\mathfrak{P}_1\mathfrak{P}_2\mathfrak{P}_3$ in $L$, I would need to know which come from a splitting of type (2) and which from type (3) in order to compute the densities in $K$.

We know that half the primes split going from $K$ to $L$, so it would seem intuitive that exactly half the primes in (3) would split completely in $L$. If this is true, then that would allow us to compute the densities in $K$. However, I don't see a way to prove this as the primes that do split going from $K$ to $L$ might be the primes appearing in the factorizations in (1) and (2).

I'm also curious if there's a way to find out precisely which primes split in which way in $K$? This was a follow-up question, but $L/\mathbb{Q}$ is not abelian and $K/\mathbb{Q}$ is not Galois, so this seems very hard unless there's some trick around it.