I have to describe lattice of subfields of $\mathbb{Q}(\zeta_{13})$.
I know that :
$$[\mathbb{Q}(\zeta_{13}):\mathbb{Q}]=\phi(13)=12$$
And $$\zeta_{13}=e^{\frac{2\pi ik}{13}} $$ with $k=0 ,...,12$
I think that $\mathbb{Q}(\sqrt13)$ of degree 2 over $\mathbb{Q}$, it is a subfield of $\mathbb{Q}(\zeta_{13})$ but I don't know how to go on...