I want to find, classify $\operatorname{Gal}(\mathbb{Q}(\zeta_n + \zeta_n^{-1})/\mathbb{Q})$ group up to isomorphism where $\zeta_n = e^{\frac{2\pi i}{n}}$. I know $\zeta_n$ is a root of $x^2 - (\zeta_n + \zeta_n^{-1})x +1$ and $[\mathbb{Q}(\zeta_n) : \mathbb{Q}(\zeta_n + \zeta_n^{-1}) ]=2$ for $n>2$.
My background of Galois theory is the very basic definition. i.e., it is an algebraic normal separable extension and if $E$ is Galois over $F$, $|Gal(E/F)|=[E:F]$.
So if I know correctly, to use the above results, I need to show $\mathbb{Q}(\zeta_n + \zeta_n^{-1})$ is Galois over $\mathbb{Q}$, what is the nice way to showing this?