irreducible polynomial of $\alpha$ over $\mathbb{Q}$ and $[\mathbb{Q}(\alpha):\mathbb{Q}]$.

Question

Let $\epsilon=\exp({\frac{2i\pi}{9}})$ and $\mathbb{Q}(\epsilon)$ be a cyclotomic extension of $\mathbb{Q}$ of order 9.

(i) Sketch the entire lattice diagram of subfields of $\mathbb{Q}(\epsilon)$.

(ii) If $\mathbb{Q}(\alpha)$ is a subfield of $\mathbb{Q}(\epsilon)$, determine the irreducible polynomial of $\alpha$ over $\mathbb{Q}$ and $[\mathbb{Q}(\alpha):\mathbb{Q}]$.