(i) Let $\zeta_{N}=\mathrm{e}^{2 \pi i / N} \in \mathbb{C}$ for $N \geqslant 1$. For the cases $N=11,13$, is it possible to express $\zeta_{N}$, starting with integers and using rational functions and (possibly nested) radicals? If it is possible, briefly explain how this is done, assuming standard facts in Galois Theory.

(ii) Let $F=\mathbb{C}(X, Y, Z)$ be the rational function field in three variables over $\mathbb{C}$, and for integers $a, b, c \geqslant 1$ let $K=\mathbb{C}\left(X^{a}, Y^{b}, Z^{c}\right)$ be the subfield of $F$ consisting of all rational functions in $X^{a}, Y^{b}, Z^{c}$ with coefficients in $\mathbb{C}$. Show that $F / K$ is Galois, and determine its Galois group. [Hint: For $\alpha, \beta, \gamma \in \mathbb{C}^{\times}$, the map $(X, Y, Z) \longmapsto(\alpha X, \beta Y, \gamma Z)$ is an automorphism of $F$.]