Let $p$ be an odd prime, $\zeta_p=e^{{2\pi i}/p}\in\mathbb{C}$, $F=\mathbb{Q}(\zeta_p)$. Let $E\subseteq\mathbb{C}$ be the splitting field of $X^p-2$ over $\mathbb{Q}$. Let $F_0=F\cap\mathbb{R}$ and $E_0=E\cap \mathbb{R}$.
(a) Show that $F\subseteq E$. Determine $\text{Gal}(F/\mathbb{Q})$ and $H=\text{Gal}(E/F)$.
(b) Describe the action of $G=\text{Gal}(E/\mathbb{Q})$ on $H$ by conjugation.
(c) Show that $F_0=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$. Is $F_0/\mathbb{Q}$ a Galois extension? If so, find its Galois group.
(d) Is $E_0/F_0$ a Galois extension? If so, find its Galois group.
(e) Determine $\text{Gal}(E/E_0)$ and $\text{Gal}(E/F_0)$.
I only have some ideas on (a).
(a) Since $E$ is the splitting field of $X^p-2$ over $\mathbb{Q}$, and $X^p-2=2\left(\left(\frac{X}{\sqrt[2]{2}}\right)^p-1\right)$, we know that $E=\mathbb{Q}(\sqrt[2]{2},\zeta_p)$, and hence $F\subseteq E$.
Let $\sigma: \zeta_p\mapsto\zeta_p^a$ be an automorphism of $F$, where $(a,p)=1$, then $\sigma\in\text{Gal}(F/\mathbb{Q})$. Since the order of $\sigma$ is equal to $\#\text{Gal}(F/\mathbb{Q})=p-1$, we have $\text{Gal}(F/\mathbb{Q})=\langle\sigma\rangle$.
Let $\tau: \zeta_p\mapsto\zeta_p, \sqrt[p]{2}\mapsto\sqrt[2]{2}\zeta_p$ be an automorphism of $E=\mathbb{Q}(\sqrt[p]{2},\zeta_p)$, then $\tau\in\text{Gal}(E/F)$. Since the order of $\tau$ in $\langle\tau\rangle$(generated by $\tau$) is $p$, which is equal to $\#\text{Gal}(E/F)$, we have $\text{Gal}(E/F)=\langle\tau\rangle$. Therefore, $H=\langle\tau\rangle$.
...
Could you please help me other problems? Thanks.