For a positive integer $N$, let $\mathbb{Q}\left(\boldsymbol{\mu}_{N}\right)$ be the cyclotomic field obtained by adjoining all $N$-th roots of unity to $\mathbb{Q}$. Let $F=\mathbb{Q}\left(\boldsymbol{\mu}_{24}\right)$.

(i) Determine the Galois group of $F$ over $\mathbb{Q}$.

(ii) Find all $N>1$ such that $\mathbb{Q}\left(\boldsymbol{\mu}_{N}\right)$ is contained in $F$.

(iii) List all quadratic and quartic extensions of $\mathbb{Q}$ which are contained in $F$, in the form $\mathbb{Q}(\alpha)$ or $\mathbb{Q}(\alpha, \beta)$. Indicate which of these fields occurred in (ii).

[Standard facts on the Galois groups of cyclotomic fields and the fundamental theorem of Galois theory may be used freely without proof.]