I've had trouble coming up with one.
I know that if I could find
an irreducible poly $p(x)$ over $\mathbb{Q}$ which has roots $\alpha, \beta, \gamma\in Q(\alpha)$,
then $|\mathbb{Q}(\alpha) : \mathbb{Q}| $ = 3 and would be a normal extension, as $\mathbb{Q}(\alpha)=\mathbb{Q}(\alpha,\beta,\gamma)$ would be a splitting field of $f$ over $\mathbb{Q}$.
However, this is a lot of conditions to find by luck...
Any help appreciated!