Finding the subfields of the cyclotomic field of order $5$ This is part of an exercise from Hungerford's Algebra:

Find all intermediate fields in the field extension $F_5/\mathbb{Q}$, where $F_5$ is the cyclotomic extension of $\mathbb{Q}$ of order $5$.

I figured out the automorphism group, it's $\mathbb{Z}_4$. So, since there is only one non-trivial subgroup of order $2$, by the Fundamental Theorem, I should find precisely one intermediate field of degree $2$, which (in my understanding) in this case means I should find an element with degree $2$ minimal polynomial. But, apparently I couldn't. I'll appreciate any help.
Also, in general, is my approach the only way to find the intermediate fields? In other words, do we always need to do a trick, and find an element with a minimal polynomial of desired degree, or is there another approach that doesn't require a separate trick in each different case? Thanks!
 A: Remember what the fundamental theorem of Galois theory says: if $L/K$ is a finite Galois extension, then given a subgroup $H\subseteq\mathrm{Gal}(L/K)$, the intermediate field corresponding to it is
$$L^H=\{\alpha\in L:\sigma(\alpha)=\alpha\text{ for all }\sigma\in\mathrm{Gal}(L/K)\}$$
Observe that it's not enough to know what $\mathrm{Gal}(L/K)$ is isomorphic to, you have to actually know what the automorphisms are (which is good advice regardless).
In your situation, it's not enough to know that $\mathrm{Gal}(F_5/\mathbb{Q})\cong\mathbb{Z}/4\mathbb{Z}$, you should know that
$$\mathrm{Gal}(F_5/\mathbb{Q})=\{\sigma_r:F_5\to F_5, \;\sigma_r(\zeta_5)=\zeta_5^r:r=1,2,3,4\}$$
You should figure out an explicit isomorphism $\mathrm{Gal}(F_5/\mathbb{Q})\cong\mathbb{Z}/4\mathbb{Z}$, i.e., write down which $\sigma_r$ goes to which element of $\mathbb{Z}/4\mathbb{Z}$. Then you'll know which $\sigma_r$ comprise the subgroup of $\mathrm{Gal}(F_5/\mathbb{Q})$ of order $2$. (Then you can get to work finding the subfield fixed by them.)
