Let $\zeta_{16}=e^{2\pi i/16}$, then we know that its minimal polynomial is $\Phi_{16}=x^8+1$. Furthermore, $$ \operatorname{Gal}(\mathbb Q(\zeta_{16})/\mathbb Q)=(\mathbb Z/16)^\times=C_2\times C_4. $$ We can choose the generators of $C_2\times C_4$: take $(1,0)$ to be $\zeta\mapsto\zeta^7$ and $(0,1)$ to be $\zeta\mapsto\zeta^3$. Here, any automorphism of $\mathbb Q(\zeta_{16})$ is determined by the image of $\zeta$.
Now I want to draw the lattice of subfields of this extension, which corresponds to the subgroup lattice of $C_2\times C_4$. Here is my result.
I have two questions:
Are these lattices correct?
I basically used brute force to search for elements that are fixed under the various subgroups and calculate their minimal polynomials, and then use the degrees of extensions to deduce that they're the correct subfield. But this method is laborious. Is there an easier algorithm?
Any help would be appreciated. Thank you in advance!