First of all, I've read some questions that have similar title, but I didn't find any answers that correspond to what I have in mind.
So, I am an undergraduate student starting my investigations in Galois Theory, and I have sort of computational questions:
Suppose we are looking for the intermediate fields of the Galois extension $\mathbb{Q}(\zeta_7)/\mathbb{Q}$, where $\zeta_7$ is a non-real root of $x^{7}=1$.
I know that the Fundamental Theorem of Galois Theory gives us the correspondence between these intermediate fields and the subgroups of the group $Gal(\mathbb{Q}(\zeta_7)/\mathbb{Q})$.
So, this group is isomorphic to $(\mathbb{Z}/7\mathbb{Z})^{\times}\cong \mathbb{Z}/6\mathbb{Z}$, which has 4 subgroups, namely: the identity, generated by 2, generated by 3 and the whole group.
Ok, from this we get that there are 2 proper subfields in the extension with degrees 2 and 3, corresponding to these proper subgroups.
Now, some thoughts occurred to me:
We know that $Gal(\mathbb{Q}(\zeta_7)/\mathbb{Q})\cong \mathbb{Z}/6\mathbb{Z}$, but we do not immediately know which automorphisms of the Galois group correspond to the elements of $\mathbb{Z}/6\mathbb{Z}$. In other words, we do not know immediately which of our known automorphisms are, say, in the proper subgroup of index 2 (information we would use to find the intermediate fields, by checking which elements of $\mathbb{Q}(\zeta_7)$ get fixed by them). Of course I could check the order, for example, of the automorphism that sends $\zeta_7\mapsto (\zeta_7)^{2}$ and find the corresponding element of $\mathbb{Z}/6\mathbb{Z}$, but, from the examples I've been doing, I know that this takes a while to do (maybe not so much in this example, but in some bigger extension it certainly does).
Isn't there any way of looking for these automorphisms in a more geometrical way? In such a way that it would be quite obvious which elements of $Gal(\mathbb{Q}(\zeta_7)/\mathbb{Q}$ correspond to which elements of $\mathbb{Z}/6\mathbb{Z}$?
By geometrical I mean, for example, looking at the unit circle and imagining such an automorphism moving things around. I've tried to think that way but I didn't get much out of it. Is this possible? If so, I believe this would help us too with the job of finding the corresponding fixed field, (which too involves long calculations) right?
Thanks in advance to all.