Let $\omega = e^{\frac{2\pi i}{5}}$
- prove that $K = \mathbb{Q}[\omega]$ is a splitting field for the polynomial $x^5-1$ over $\mathbb{Q}$
Determine the Galois group $G = Gal(K/\mathbb{Q})$
For each subgroup $H$ of $G$ (including $G$ and {Id}) describe explicitly the fixed field $K^H$ and determine $[K^H : \mathbb{Q}]$ and give a $\mathbb{Q}$-basis for $K^H$.
I have that all the fifth roots of unity are powers of $\omega$, thus $K = \mathbb{Q}[\omega]$ is a splitting field. Is this sufficient? As for the rest, I do not have much practice determining the Galois groups. Explicit direction would be much appreciated, as this is a guiding example for the rest of my work.