Explanation

(a)
Given equation is
$\alpha^{2} + \alpha + 1 = 0$
$\therefore\ \ \ \alpha = \frac{- 1 \pm \sqrt{1 - 4}}{2} = \frac{- 1 \pm \sqrt{3}i}{2}$
Let $\alpha = \omega,\ \omega^{2}$
If $\alpha = \omega$, then
$\alpha^{31} = (\omega)^{31} = \omega = \alpha$
If $\alpha = \omega^{2}$ , then
$\alpha^{31} = \left( \omega^{2} \right)^{31} = \omega^{62} = \omega^{2} = \alpha$
Hence, $\alpha^{31}$ is equal to $\alpha$