Is there a Chebotarev‘s theorem for non-Galois extension over Q？

Question

For a Galois extension $K/\mathbb{Q}$, the Chebotarev Density Theorem predicts the density of primes with a certain splitting type.

I'm wondering if there is a similar result for non-Galois extension?

Any references are welcome!

Answer 1

Actually the usual Chebotarev density theorem in the Galois case can also be applied to the non-Galois case.

For example, consider a non-Galois cubic extension $K=\mathbb{Q}[x]/(f)$. I claim that the following splitting types of unramified primes $p$ occur with the following densities $\delta$: $$\delta(1,1,1) = 1/6, \quad \delta(1,2) = 1/2 , \quad \delta(3) = 1/3.$$ To see this, let $L$ be the Galois closure of $K$, which is an $S_3$-extension of $\mathbb{Q}$. The Galois group $S_3$ acts on the roots of $f$ and the splitting type is determined by the action of the Frobenius element $\mathrm{Frob}_p$.

We obtain splitting type $(1,1,1)$ (completely split) if and only if $\mathrm{Frob}_p$ fixes all the roots, which means exactly that $\mathrm{Frob}_p$ is trivial. So Chebotarev gives the expected $1/6$. We obtain splitting type $(1,2)$ if and only if $\mathrm{Frob}_p$ is a transposition in $S_3$, of which there are $3$, so we get density $3/6 = 2/3$. Finally we obtain splitting type $(3)$ if and only if $\mathrm{Frob}_p$ is a $3$-cycle in $S_3$, of which there are $2$, giving density $2/6 = 1/3$, as claimed.

In general, a splitting type corresponds to some conjugacy invariant subset of the Galois group of the splitting field, as can be seen by considering the action on the roots of the defining polynomial. The density of this is then calculated using the usual Chebotarev for Galois extensions.