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

For a set $S$ of prime ideals, its Dirichlet density iswhen this limit exists.

Let $G=\mathrm{Cl}(K)$ and let $\hat{G}$ be its character group. Character orthogonality givesFor $\mathrm{Re}s>1$, the prime term of $\log L(s,\chi )$ isup to a function bounded as $s\to 1^{+}$, since higher prime powers converge there. The trivial character contributeswhereas every nontrivial character contributes $O(1)$ because its $L$-function is nonzero at $1$. Thereforeand the density is $\boxed{1/h_K}$.

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

PutIt contains $K=\mathbb{Q}(\sqrt{26})$ and has degree two over $K$. The three quadratic subfields have discriminants $8$, $13$, and $104$, so the biquadratic discriminant formula givesThe relative discriminant formulatherefore gives ${\mathfrak{d}}_{H/K}={\mathcal{O}}_K$: no finite prime ramifies. The extension is totally real, so no infinite prime ramifies either. Since $h_K=2$, the Hilbert class field has degree two over $K$. Consequently

A prime ideal of $K$ is principal exactly when its Artin symbol inis the identity. The group has order two. Applying the Chebotarev density theorem to the identity conjugacy class gives density