All texts I could find about Tchebotarev theorem prove it for a general Galois extension $L|K$.
I wonder if there is a specific proof for the simpler case of $K=\mathbb{Q}$, i.e.:
For a given conjugacy class $C$ of $\text{Gal}(L|\mathbb{Q})$, the density of the set of unramified primes $p\in\mathbb{Z}$ with $\text{Frob}_{\mathfrak{P}}\in C$ for all $\mathfrak{P}\mid p$ is equal to $\frac{\#C}{\#\text{Gal}(L|\mathbb{Q})}$.
I imagine the proof should be more elementary, but I don't know if that's the case.