## Decide whether a field extension contains a given element

Let $\omega = \frac{-1}{2} + \frac{\sqrt{3}}{2}i$. Decide whether or not $\sqrt{3} \in \mathbb{Q}(i,\omega)$.

Evidently, $i(\omega^2 - \omega) = \sqrt{3}$; so $\sqrt{3} \in \mathbb{Q}(i,\omega)$.

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