Roots of negative unity

$\large x^{2n} + x^{2n-1} + x^{2n-2} + \dots + x^2 + x+ 1$

In the polynomial above, $n$ is a positive integer. The roots of this polynomial can be expressed as $(-1)^{\frac{m}{\alpha n + \beta}+m}$ where $m$ is an integer that is never equal to $k(\alpha n + \beta)$ where $k$ is odd, $\alpha$ and $\beta$ are positive integers where $\beta < \alpha < 8$ .

Find $\cfrac{\alpha + \beta}{\alpha \beta}$