# Math Help - galois group

1. ## galois group

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

2. Originally Posted by apple2009
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: $\text{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q}) \cong (\mathbb{Z}/7\mathbb{Z})^\times \cong \mathbb{Z}/6\mathbb{Z}$.

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

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

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

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

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