I have this two exercise where I have some problems:
(a) We consider the finite field extension $E := \mathbb{Q}( \sqrt[4]{2}, i)$ over $\mathbb{Q}$. Show that $E$ is a Galois extension of $\mathbb{Q}$ and determine the isomorphism type of the Galois group $G := Gal(E/ \mathbb{Q})$.
(b) For each subgroup $H ≤ G$, determine the field extension $Q ⊆ L ⊆ E$ corresponding to the main theorem of Galois theory. For which of these intermediate fields $L$, is $L$ a Galois extension of $\mathbb{Q}$?
So for a) I have that since the finite extension field is normal and separable we have a Galois extension field.
For the isomorphism type I need to find $Gal(E/ \mathbb{Q})$ first. I hope it is clear that $[\mathbb{Q}( \sqrt[4]{2}, i):\mathbb{Q}]=8$. So if we find 8 $E$-Automorphisms over $\mathbb{Q}$ we have found our $Gal(E/ \mathbb{Q})$.
The 8 Automorphisms that I have in mind are
$\tau_1:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow \sqrt[4]{2}\\ i \rightarrow i \end{matrix}\right. , \tau_2:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow \sqrt[4]{2}\\ i \rightarrow -i \end{matrix}\right.,\\\tau_3:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow -\sqrt[4]{2}\\ i \rightarrow i \end{matrix}\right.,\tau_4:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow -\sqrt[4]{2}\\ i \rightarrow -i \end{matrix}\right., \\ \tau_5:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow i\sqrt[4]{2}\\ i \rightarrow i \end{matrix}\right.,\tau_6:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow -i\sqrt[4]{2}\\ i \rightarrow i \end{matrix}\right.,\\ \tau_7:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow i\sqrt[4]{2}\\ i \rightarrow -i \end{matrix}\right.,\tau_8:\left\{\begin{matrix} \sqrt[4]{2} \rightarrow -i\sqrt[4]{2}\\ i \rightarrow -i \end{matrix}\right.$
Therefore we have that $G =\{id,\tau_2,\tau_3,\tau_4,\tau_5,\tau_6,\tau_7,\tau_8 \}$. So now I had in mind to check the order of this 8 Automorphisms to obtain the isomorphism type. for $\tau_2, \tau_3, \tau_4$ I have order $2$ but for $\tau_5,\tau_6,\tau_7,\tau_8$ I dont know how to find the order. Can someone help me? Even once I found the order of the other 4, I wouldn't know how to find the type of isomorphy. could someone help me?
While for b) I think (Sorry for the confusion but I am new to the Galois theory)I have to take some of the automorphisms of $G$ and from them I have to obtain the intermediate fields but here too I don't know how to do it.