What are the subfields of $\mathbb{Q}(\zeta_7)$

Question

Let $\zeta = e^{\frac{2i\pi}{7}}$. I know that the automorphisms of $\mathbb{Q}(\zeta)$ are isomorphic to the cyclic group with $6$ elements so that the subfields of $\mathbb{Q}(\zeta)$ correspond to the subgroups of $C_6$. I found one of the proper subfields is $\mathbb{Q}(\zeta + \zeta^2 + \zeta^4)$. I was able to determine this by observing the form of the elements that are fixed by the mapping that takes $\zeta$ to $\zeta^2$. How can I determine the other subfield that corresponds to the group of $2$ automorphisms consisting of the trivial mapping and the mapping that sends $\zeta$ to $\zeta^6$?