Say we have the polynomial $x^4-2$, the splitting field of this over $\Bbb Q$ is $\Bbb Q(\alpha, i)$, $\alpha=\sqrt[4]{2}$, and its Galois group is isomorphic to $D_8$.
Now I know a way to find the lattice of subfields of $\Bbb Q(\alpha, i)$ in the Galois correspondence.
If we know the subgroup structure of $D_8$ then we can find the corresponding subfield for each subgroup.
Say for example $h$ is an automorphism which maps $\alpha \rightarrow i\alpha$ and $i \rightarrow i$ and another $g$ which maps $\alpha \rightarrow \alpha$ and $i \rightarrow -i$. Then these generate the galois group and a subgroup is $<g,h^2>$ which has as its fixed field the intersection of the fixed fields of $g$ and $h^2$. Thos in turn are found by noting that g has order 2, so its fixed field must be degree 8/2=4 over Q and it must contain $\Bbb Q(\alpha) $ therefore it must be $\Bbb Q(\alpha)$, similarly $h^2$ has fixed field $\Bbb Q(i,\sqrt{2})$, the intersection (fixed field of $g, h^2$) is then $\Bbb Q(\sqrt{2})$
This method works just fine but it requires that you fully know the subgroup structure of the Galois group in order to find the subfields of the galois extension. My questions are
1)Is there a way to find the subfields without knowing anything about the structure of the group
2) Can we then work in the other direction and construct the subgroups from the lattice of subfields.
If I haven't quite asked the right question here but you know of some way of doing these problems without having to know subgroup structures , please feel free to answer anyway.