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Let $K$ be cyclotomic field generated over $\Bbb Q$ by the $9$th root of unity $z$, having Galois group $G$. Show that it is a cyclic extension of degree $6$ of $\Bbb Q$ and by making use of the subgroup $H$ of index $3$ of $G$ construct a Galois extension $K'$ of degree $3$. Show that $K'=\Bbb Q [\cos \frac {2 \pi} q]$ and determine the minimal polynomial of $2 \cos \frac {2 \pi} q$ over $\Bbb Q$.

I need help to solve this.

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Let $\omega$ be a primitive nineth root of unity. The Galois group $G$ is given by $\{\sigma_k \mid k \in (\mathbb{Z}/9\mathbb{Z})^\times\}$ where $\sigma_k (\omega ) = \omega^k$, so $G \cong (\mathbb{Z}/9\mathbb{Z})^\times \cong \mathbb{Z}/6\mathbb{Z}$. The only subgroup $H$ of index $2$ consists of $\sigma_1 = \mathrm{id}$ and $\sigma_{-1} = \sigma_8$. Let $K^H$ be the fixed field of $H$ and consider the elemtent $\theta = \omega + \omega^{-1}= \omega + \omega^8$. Then $[K^H:\mathbb{Q}]=3$ and $\theta$ is fixed by both elements, so $\theta \in K^H$. if you can show that $\theta \notin \mathbb{Q}$, then $K^H = \mathbb{Q}(\theta)$, since $3$ is prime.

For the minimal polynomial of $\theta = 2 \cos(\frac{2\pi}{9})$, which then has degree 3, express $1$, $\theta$, $\theta^2$ and $\theta^3$ as powers of $\omega$ and try to spot a relation!

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