Determine the Galois group over Q of the splitting field of the question. List all of the subgroups of the Galois group. List all of the subfield of the splitting field of the equation.

(X^7)-1

Hint: \(\displaystyle \text{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q}) \cong (\mathbb{Z}/7\mathbb{Z})^\times \cong \mathbb{Z}/6\mathbb{Z}\).

Find the subgroups of it and correspond them to the intermediate fields.

For instance, if you choose a generator of the above group as \(\displaystyle \sigma:\zeta_7 \mapsto \zeta_7^3\), then one of the subgroups of it is \(\displaystyle \{1, \sigma^3\}\).

The reason why I choose the above generator (among them) is because 3 has order 6 in \(\displaystyle (\mathbb{Z}/7\mathbb{Z})^\times\) ( Meanwhile, 2 has order 3 in \(\displaystyle (\mathbb{Z}/7\mathbb{Z})^\times\)).

Since \(\displaystyle \zeta_7 + \sigma^3\zeta_7=\zeta_7 + \zeta_7^{3^3}=\zeta_7 + \zeta_7^{-1}\), the corresponding intermediate field is \(\displaystyle \mathbb{Q}(\zeta_7+\zeta_7^{-1})\).

I'll leave it to you to find the remaining subgroups of \(\displaystyle <\sigma>\) and their corresponding intermediate fields.