Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 3 / f / Solution

For $K=\mathbb{Q}(\sqrt{-7})$,The character is odd, ${\chi }_3(2)=1$, andThe supplied odd-character formula gives

The analytic class number formula for this imaginary quadratic field isbecause $w_K=2$ and $d_K=-7$. Since ${\mathrm{Res}}_{s=1}\zeta (s)=1$, comparison with $L(1,{\chi }_3)$ yields $\boxed{h_K=1}$.