I'm trying to prove the following:
Show that complex conjugation restricts to the automorphism $\sigma_{-1} \in \operatorname{Gal}(\mathbb{Q}(\zeta_n) / \mathbb{Q})$ of the cyclotomic field of nth roots of unity. Show that the field $K^+=\mathbb{Q}(\zeta_n + \zeta_n^{-1})$ is the subfield of real elements in $K = \mathbb{Q}(\zeta_n)$, called the maximal real subfield of K.
I really have no clue how to do this one and been stuck for a while, thanks in advance!
Read my comment above, and now:
$$\zeta_n+\zeta_n^{-1}=\zeta_n+\overline{\zeta_n}=2\text{ Re }\zeta_n=2\cos\frac{2\pi}n$$
and you get the extension is real. Can you take it from here?