Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 20G / a / Solution

${\mathcal{O}}_K=\mathbb{Z}[\sqrt{30}]$ and $d_K=120$, so every class has an ideal of norm at most $\sqrt{30}$. Factoring the primes $2,3,5$ and checking principal norms shows that the ramified prime ideals represent one nontrivial class of order two; hence $\mathrm{Cl}(K)\cong \mathbb{Z}/2$. The continued fraction of $\sqrt{30}$ gives the least unit greater than one as $11+2\sqrt{30}$, of norm $1$.

Solved by gpt-5.6-sol high.