Find the irreducible polynomial of $\zeta_{12}$ and $\zeta_9$ over the field $Q(\zeta_3)$.

Question

Let $\zeta_n=exp(2 \pi i/n)$ Find the irreducible polynomial of $\zeta_{12}$ and $\zeta_9$ over the field $Q(\zeta_3)$.

Here I can find polynomial satisfied by these elements over $Q(\zeta_3)$ but i am unable to prove that these polynomial will be irreducible over $Q(\zeta_3)$.For example $\zeta_{12}$ satisfies polynomial $x^2-exp(\pi/3)$ over $Q(\zeta_3)$ but i am unable to prove that this polynomial will be irreducible over $Q(\zeta_3)$.Give some general idea!

Answer 1

Using Galois theory; Let $3|n$.

The irreducible polynomials of $\zeta_n$ over $\mathbf{Q}(\zeta_3)$ is the product $\prod(X-\sigma(\zeta_n))$ (the product is over $\sigma \in G_n =\mathrm{Gal}(\mathbf{Q}(\zeta_n)/\mathbf{Q}(\zeta_3))$), and $G_n$ can be calculated, easily, $\{ \sigma | \sigma (\zeta_n) = {\zeta_n}^i\ ( i\in \ker((\mathbf{Z}/n\mathbf{Z})^{\times} \rightarrow (\mathbf{Z}/3\mathbf{Z})^{\times}))\}$.