Galois group of intermediate fields Find all subfields $M \subset \mathbb{Q}(\zeta_{7})$ where $\zeta_7=e^{2\pi i/7}$ and determine $Gal(\mathbb{Q}/M)$ and $Gal(M/\mathbb{Q})$.
I've found that there are 2 intermediate fields
$L=\mathbb{Q}( \zeta_7 + \zeta_7^6)$ and $M=\mathbb{Q}( \zeta_7 + \zeta_7^2+ \zeta_7^4)$.
I've also found that $Gal(\mathbb{Q}(\zeta_7)/M)=\{\sigma_1, \sigma_2, \sigma_4\}$ and $Gal(\mathbb{Q}(\zeta_7)/L)=\{\sigma_1, \sigma_6 \}$ where $\sigma_a(\zeta_7)=\zeta_7^a$. I just need to find $Gal(M/\mathbb{Q})$ and $Gal(L/\mathbb{Q})$
 A: You've already seen that $\text{Gal}({\mathbb Q}(\zeta_7)/{\mathbb Q}) \cong C_6$. You've also seen that $\text{Gal}({\mathbb Q}(\zeta_7)/M) \cong C_3$ and that $\text{Gal}({\mathbb Q}(\zeta_7)/L) \cong C_2$. By Galois Correspondence, $\text{Gal}(M/{\mathbb Q}) \cong C_6 / C_3 \cong C_2$ and $\text{Gal}(L/{\mathbb Q}) \cong C_6/C_2 \cong C_3$.
If you want these Galois groups explicitly, as groups of automorpmisms of the field extensions $M/{\mathbb Q}$ and $L/{\mathbb Q}$, note that in both cases you just have to find a single non-trivial automorphism; that will automatically generate the Galois group.
For $M/{\mathbb Q}$, the map sending $\zeta_7$ to $\zeta_7^6$ (i.e., $\sigma_6$ restricted to $M$) is the generator; note that it indeed leaves $\zeta_7 + \zeta_7^6$ invariant. For $L/{\mathbb Q}$, the map sending $\zeta_7$ to $\zeta_7^2$ (i.e., $\sigma_2$ restricted to $L$) is a generator; this one indeed leaves $\zeta_7 + \zeta_7^2 + \zeta_7^4$ invariant. The other one is $\sigma_4$ restricted to $L$.
