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
$$
\boxed{\frac12}.
$$

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-123/5/e/solution