Let $\zeta=e^{2 \pi i / 5}$ and let $K=\mathbb{Q}(\zeta)$. Show that the discriminant of $K$ is 125 . Hence prove that the ideals in $K$ are all principal.

Verify that $\left(1-\zeta^{n}\right) /(1-\zeta)$ is a unit in $K$ for each integer $n$ with $1 \leqslant n \leqslant 4$. Deduce that $5 /(1-\zeta)^{4}$ is a unit in $K$. Hence show that the ideal $[1-\zeta]$ is prime and totally ramified in $K$. Indicate briefly why there are no other ramified prime ideals in $K$.

[It can be assumed that $\zeta, \zeta^{2}, \zeta^{3}, \zeta^{4}$ is an integral basis for $K$ and that the Minkowski constant for $K$ is $3 /\left(2 \pi^{2}\right)$.]