Find the Galois group $G = \text{Gal}(\mathbb{Q}(\omega_{12})/\mathbb{Q})$, and its lattice of subgroups, where $\omega_{12}$ is the 12th root of unity.
We have that $$\text{Gal}(\mathbb{Q}(\omega_{12})/\mathbb{Q}) \cong (\mathbb{Z}/12\mathbb{Z})^{\times},$$ where $(\mathbb{Z}/12\mathbb{Z})^{\times}$ is the multiplicative group of units. And we know that the units of $(\mathbb{Z}/n\mathbb{Z})$ are the generators of $(\mathbb{Z}/n\mathbb{Z})^{+}$. These are the elements that are coprime to 12. Therefore, $(\mathbb{Z}/12\mathbb{Z})^{\times} = \{1,5,7,11 \}$. The proper subgroups of $(\mathbb{Z}/12\mathbb{Z})^{\times}$ are $\{1 \}, \{1,5 \}, \{1,7 \}, \{1, 11 \}$.
The group $(\mathbb{Z}/12\mathbb{Z})^{\times}$ fixes $\mathbb{Q}$, $\{1,5\}$ fixes $\mathbb{Q}(\omega_{12}^5)$, $\{1,7 \}$ fixes $\mathbb{Q}(\omega_{12}^7)$, $\{1,11 \}$ fixes $\mathbb{Q}(\omega_{12}^{11})$ and $\{1 \}$ fixes $\mathbb{Q}(\omega_{12})$.
Also, is $\{1, \omega, \omega^2, ..., \omega^9, \omega^{10} \}$ a $\mathbb{Q}$ basis of $\mathbb{Q}(\omega_{12})$?