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