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

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