How to show $\operatorname{Gal}(\Bbb F_2(\alpha)/\Bbb F_2)$ = {$e, \sigma, \sigma^2$}

Question

How to show $\operatorname{Gal}(\Bbb F_2(\alpha)/\Bbb F_2)=\{e, \sigma, \sigma^2\}$? $α$ is a root lying in an extension of $\Bbb F_2$ where $\sigma(\alpha) = \alpha^2$

I have $\sigma^2(\alpha) = \alpha^4$ and $\sigma^3(\alpha) = \alpha^8$ but I'm not sure where to go further.

Also how do you prove that $\alpha^2 + \alpha$ and $ \alpha$ are conjugate over $\Bbb F_2$?

I'm not sure where to start