# galois group

#### 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

#### TheArtofSymmetry

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.

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