Past exam of the number theory course of the University of Oxford / 2011 / B / 6 / c / Solution

For degree $3$ there are either three real embeddings or one real embedding and one complex pair. The respective Minkowski bound for ideal classes areUnder the given discriminant bound they are at most $\frac{2}{9}\sqrt{45}<2$ and $\frac{8}{9\pi }\sqrt{45}<2$. Hence every ideal class contains an integral ideal with positive integer norm strictly below $2$, so with norm $1$. The only such ideal is ${\mathcal{O}}_L$. Therefore