Prove $\operatorname{Gal}(\mathbb{Q}(e^{\pi i/q}):\mathbb{Q}) \cong C_{q-1}$ for odd prime q.

Question

Let $u=e^{\pi i/q}$, where $q$ is an odd prime number. I want to Show that $\operatorname{Gal}l(\mathbb{Q}(u):\mathbb{Q})\simeq C_{q-1}$

My thought is to find the polynomial $x^q+1$ and find a minimal polynomial for $u$ so that I can find its other roots, but I'm unsure of how to do this.

Any ideas?