Let $m$ be an integer greater than 1 and let $\zeta_{m}$ denote a primitive $m$-th root of unity in $\mathbb{C}$. Let $\mathcal{O}$ be the ring of integers of $\mathbb{Q}\left(\zeta_{m}\right)$. If $p$ is a prime number with $(p, m)=1$, outline the proof that

$$p \mathcal{O}=\wp_{1} \cdots \wp_{r},$$

where the $\wp_{i}$ are distinct prime ideals of $\mathcal{O}$, and $r=\varphi(m) / f$ with $f$ the least integer $\geqslant 1$ such that $p^{f} \equiv 1 \bmod m$. [Here $\varphi(m)$ is the Euler $\varphi$-function of $\left.m\right]$.

Determine the factorisations of $2,3,5$ and 11 in $\mathbb{Q}\left(\zeta_{5}\right)$. For each integer $n \geqslant 1$, prove that, in the ring of integers of $\mathbb{Q}\left(\zeta_{5^{n}}\right)$, there is a unique prime ideal dividing 2 , and a unique prime ideal dividing 3 .