Let $K$ be a field of characteristic 0 . It is known that soluble extensions of $K$ are contained in a succession of cyclotomic and Kummer extensions. We will refine this statement.

Let $n$ be a positive integer. The $n$-th cyclotomic field over a field $K$ is denoted by $K\left(\boldsymbol{\mu}_{n}\right)$. Let $\zeta_{n}$ be a primitive $n$-th root of unity in $K\left(\boldsymbol{\mu}_{n}\right)$.

(i) Write $\zeta_{3} \in \mathbb{Q}\left(\boldsymbol{\mu}_{3}\right), \zeta_{5} \in \mathbb{Q}\left(\boldsymbol{\mu}_{5}\right)$ in terms of radicals. Write $\mathbb{Q}\left(\boldsymbol{\mu}_{3}\right) / \mathbb{Q}$ and $\mathbb{Q}\left(\boldsymbol{\mu}_{5}\right) / \mathbb{Q}$ as a succession of Kummer extensions.

(ii) Let $n>1$, and $F:=K\left(\zeta_{1}, \zeta_{2}, \ldots, \zeta_{n-1}\right)$. Show that $F\left(\boldsymbol{\mu}_{n}\right) / F$ can be written as a succession of Kummer extensions, using the structure theorem of finite abelian groups (in other words, roots of unity can be written in terms of radicals). Show that every soluble extension of $K$ is contained in a succession of Kummer extensions.