Decomposition field over Q

**roporte** · November 5th 2008, 08:00 AM

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!

**clic-clac** · November 5th 2008, 12:04 PM

$X^{4}+X^{2}+1=(X^{2}+X+1)(X^2-X+1)$
$\omega$ 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 $\mathbb{Q}$

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