Let $\alpha$ be a root of the polynomial $X^3-X-1$. The polynomial has discriminant $-23$ which is not a square, so the splitting field must have Galois group $S_3$. I would like to figure out the splitting of primes in $\mathbb{Q}(\alpha)$. The ramified case is trivial, so assume that $p$ is not ramified. By looking at possible splittings of a prime $p$ of $\mathbb{Q}$ in the Galois closure and then looking at which splittings in $\mathbb{Q}(\alpha)$ give rise to which splittings in the Galois closure, we get the following options for splittings in $\mathbb{Q}(\alpha)$:
$(p)$ is stays inert. Density: 1/3.
$(p)=\mathfrak{p}\mathfrak{q}$. Density: 1/2.
$(p)$ splits. Density: 1/6.
An old qual question I found online asks for which primes $p$ in $\mathbb{Q}$ give rise to which splitting, so to me this seems like the question asks for explicit conditions on $p$ which would tell us how it splits. This is just from a transcript written by the student, so I have no idea whether or not an answer was actually expected or if this was a trick question.
The way I've understood the norm limitation theorem of class field theory is that we can only expect to give congruence conditions in $\mathbb{Q}$ for how primes split in an abelian extension. Thus, for any nonabelian extension there is no way to express the condition in the form
$p\equiv a_1,\ldots,a_k\,(\textrm{mod } n)$
since any congruences would only give us information about splittings in the maximal abelian subextension. In this case knowing splitings in $\mathbb{Q}(\sqrt{-23})$ does not let us distinguish between splittings in$\mathbb{Q}(\alpha)$. Can anyone tell me if my intuition is right or if there actually are ways to write down explicit conditions on the primes?