Find all sub-fields of $K:=\mathbb{Q}(\sqrt[4]{5},i)$ by giving generators.
I have shown that $K/\mathbb{Q}$ is a Galois extension and that its Galois group $G:=\text{Gal}(K/\mathbb{Q})$ is generated by $\tau$ and $\sigma$, where $\tau(i)=-i, \tau\left(\sqrt[4]{5}\right)=\sqrt[4]{5}$ and $\sigma(i)=i, \sigma\left(\sqrt[4]{5}\right)=-i\sqrt[4]{5}$. Hence, $G\cong D_4$, where $D_4$ denotes the dihedral-group of $8$ elements.
Thus, there are $9$ sub-fields of $K$ (as $D_4$ has $9$ sub-groups), two of which are obviously $K$ and $\mathbb{Q}$. Now I don't know how to find the others. For example, what is the sub-field $K_1$ respectively $K_2$ generated by the subgroups $H_1:={\langle \sigma\rangle}$ respectively $H_2:={\langle \sigma^2\rangle}$? I know that $[K_1:\mathbb{Q}]=2$ and $[K_2:\mathbb{Q}]=4$, but don't know how to find elements that generate $K_1, K_2$.