We have $\xi_{12}$ denotes the 12-th root of unity. I'm having trouble getting the idea of how to find these subgroups. I know there has to be 4 since irr($\xi_{12}, \mathbb{Q}) = \Phi_{12}(X)$ which has degree 4. I also know that one of the elements has to be complex conjugation, one the identity map. When i have the third one, the 4th one has to be the combination of that one and complex conjugation, but how do i find the third one? I know Gal($\mathbb{Q}(\xi_{12})/\mathbb{Q}) \cong \mathbb{Z}_2 \times \mathbb{Z}_2$, can that maybe help me?
2026-04-02 10:09:22.1775124562
Subgroups of Gal($\mathbb{Q}(\xi_{12})/\mathbb{Q})$
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