# Math Help - Decomposition field over Q

1. ## Decomposition field over Q

Let $f = X^4 + X^2 + 1 \in \mathbb{Q}[X]$, and $\omega = \frac{1}{2}(-1+i\sqrt{3}) \in \mathbb{C}$
Prove that $\mathbb{Q}(\omega)$ is a decomposition field of $f$ over $\mathbb{Q}$.

Thanks!

2. $X^{4}+X^{2}+1=(X^{2}+X+1)(X^2-X+1)$
$\omega$ and ${\omega}^2$ are the roots of $(X^{2}+X+1)$. The roots of $(X^2-X+1)$ are $\omega '$ and ${\omega '}^2$ where $\omega '=\frac{1}{2}(1-i\sqrt3)$
As $\mathbb{Q}(\omega )$ contains $\omega, \omega, {\omega}^2, {\omega '}^2$, it is a decomposition field of $f$ over $\mathbb{Q}$