(1) Let $F=\mathbb{Q}(\sqrt[3]{5}, \sqrt{5}, i)$. What is the degree of $F / \mathbb{Q}$ ? Justify your answer.

(2) Let $F$ be a splitting field of $X^{4}-5$ over $\mathbb{Q}$. Determine the Galois group $\operatorname{Gal}(F / \mathbb{Q})$. Determine all the subextensions of $F / \mathbb{Q}$, expressing each in the form $\mathbb{Q}(x)$ or $\mathbb{Q}(x, y)$ for some $x, y \in F$.

[Hint: If an automorphism $\rho$ of a field $X$ has order 2 , then for every $x \in X$ the element $x+\rho(x)$ is fixed by $\rho$.]