 Solution

The orbit polynomial ${\prod }_{\beta \in G\alpha }(X-\beta )$ has coefficients fixed by $G$, lies in $K[X]$, and has distinct roots. Thus $\alpha$ is separable and its degree is the orbit size $|G:G_{\alpha }|$, which divides $|G|$. If $L=K(\alpha )$ and the action is faithful, only the identity fixes $\alpha$, so the degree is $|G|$.