I am trying to work out whether the field extension $\mathbb{Q}(\alpha )/\mathbb{Q}$ is radical where $\alpha =\cos (\pi /7)$. I know that $\omega +1/\omega =2\cos (\pi/7 )$ where $\omega = \exp (2\pi i/7)$ and that $\mathbb{Q}(\omega )/\mathbb{Q}$ is a radical extension since $\omega $ is a root of $x^7-1 \in \mathbb{Q}[x] $.
I suspect it is actually a radical extension but can't see a way to show it?