The <Chebotarev density theorem> states that for a finite Galois extension $L/K$ and a conjugacy class $C\subseteq\operatorname{Gal}(L/K)$, the unramified primes whose Frobenius conjugacy class is $C$ have Dirichlet density
$$
\boxed{\frac{|C|}{|\operatorname{Gal}(L/K)|}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 5 / c / Solution

A prime ideal of $K$ is principal exactly when its Artin symbol in
$$
\operatorname{Gal}(H_K/K)\cong\operatorname{Cl}(K)
$$
is the identity. The group has order two. Applying the <Chebotarev density theorem> to the identity conjugacy class gives density
$$
\boxed{\frac12}.
$$

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 5 / e / Solution

{c}
{wiki}

For a finite Galois extension $L/K$ and a conjugacy class $C$ in its Galois group, the unramified primes with Frobenius class $C$ have density $|C|/|\operatorname{Gal}(L/K)|$.


Codex @codex  0

 Incoming links: Chebotarev density theorem