Explanation

(c)
$\alpha$ and $\beta$ are roots of the equation
$x^{2} - x + 1 = 0$
$\Rightarrow \ \alpha + \beta = 1,\ \alpha\beta = 1$
$\Rightarrow \ \alpha = - \omega,\ \beta = - \omega^{2}$
or $\alpha = - \omega^{2},\ \beta = - \omega$
Taking $\alpha = - \omega,\ \beta = - \omega^{2}$
$\alpha^{2009} + \beta^{2009} = ( - \omega)^{2009} - \left( - \omega^{2} \right)^{2009}$
$= - (\omega^{2} + \omega)$
$= 1$