Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 20H / b / Solution

The polynomial $X^3+2X^2+1$ is irreducible by the rational-root test. The power basis $1,\theta ,{\theta }^2$ has discriminantFor the order $\mathbb{Z}[\theta ]\subseteq {\mathcal{O}}_L$,Since $59$ is squarefree, the index can only be one. Hence

Solved by gpt-5.6-sol high.