Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 123 / 3

For a set $S$ of nonzero prime ideals of $K$, its Dirichlet density isprovided the limit exists.

The Chebotarev density theorem says that if $L/K$ is a finite Galois extension with group $G$ and $C\subseteq G$ is a conjugacy class, then the unramified primes $\mathfrak{p}$ whose Frobenius conjugacy class equals $C$ have Dirichlet density

Let ${\mathfrak{f}}_{L/K}$ be the conductor of an abelian extension and let $\mathfrak{m}$ be divisible by it. For any $\sigma \in \mathrm{Gal}(L/K)$, the Chebotarev density theorem supplies infinitely many unramified prime ideals $\mathfrak{p}$ withOnly finitely many primes divide ${\mathfrak{m}}_0$, so $\mathfrak{p}$ may be chosen prime to ${\mathfrak{m}}_0$. It then belongs to $I_K(\mathfrak{m})$ and its image under the Artin reciprocity map is $\sigma$. Hence the Artin map is surjective.

A prime splits completely in $L$ exactly when its Frobenius conjugacy class is $\{1\}$. Applying the Chebotarev density theorem to the identity class gives

By Question 1, a prime ideal of $K$ is principal exactly when it splits completely in the Hilbert class field $H_K$. Question 3(4) therefore gives