For the following polynomials (a) $f(x)=x^4-5x^2+6, F = \mathbb{Z}_7$, (b) $g(x)=x^3-3 , F=\mathbb{Q}$ in the given field $F$ find:
i. Its decomposition field $K$
ii. The Galois group $G = Gal (K/F)$
iii. The subgroup latice of $G$ and, for each subgroup $H$ of $G$, find the subfield of $K$ that is left fixed by $H$
iv. Find the latitude of subfields of $K$ containing $F$. Try to find those that are left fixed by some subgroup of $G$, establishing the correspondence between subgroups and subfield. (Diagrams can be helped of latices). Explicitly state which of the intermediate fields are normal extensions of $F$.
I have tried to make this problem and I do not know if what I have got is fine:
i. For $f$ is $\mathbb{Q}(\sqrt{2},\sqrt{3})$ and for $g$ is $\mathbb{Q}(\sqrt[3]{3},\omega)$ where $\omega=e^{2i\pi/3}$.
ii. $Gal((\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Z}_7)\cong C_2\times C_2 $ and $Gal(\mathbb{Q}(\sqrt[3]{3},\omega)/\mathbb{Q})\cong D_3$
iii. I do not know what are the subgroups of these groups, could anyone help me with this please? In order to find the fixed field I need the subgroups but I do not know how to find them
iv. I know that each subgroup of the group of galois has a subfield of the field extension, and these subfields are the fixed field of the subgroups of the galois group, but for this I need to find the subgroups of each group of galois.