Let $\zeta$ be a primitive 12th-root of unity. How many field extensions are there between $\mathbb{Q}(\zeta^3)$ and $\mathbb{Q}(\zeta)$?
Here is my solution:
$\mathbb{Q}(\zeta)$ is the splitting field of $X^{12}-1$, so $\mathbb{Q}(\zeta):\mathbb{Q}$ is a Galois extension. The minimal polynomial of $\zeta$ is the cyclotomic polynomial $\Phi_{12}$, whose degree is 4, therefore $G(\mathbb{Q}(\zeta):\mathbb{Q})=\mathbb{Z}_{12}^*$. Then, every non trivial subgroup has order 2, so every intermediate extension of $\mathbb{Q}(\zeta):\mathbb{Q}$ is quadratic, which shows that there is no intermediate field extensions between $\mathbb{Q}(\zeta^3)$ et $\mathbb{Q}(\zeta)$.
I would like to know if my reasoning is right.
Thanks in advance.