Let $G$ be a finite group of automorphisms acting on a field $L$, with fixed field $K$.
My notes say "Let $ \alpha \in L $. Consider the set $ \{\sigma(\alpha)| \sigma \in G\}$ and suppose its distinct elements are $\alpha = \alpha_1, \alpha_2, ... \alpha_r $. Let $g = \Pi (X-\alpha_i).$ Then $g$ is invariant under $G$, since its linear factors are just permuted by elements of $G$"
I don't understand that last sentence. Why must an element of $G$ permute the linear factors? Why must an $\alpha_i$ be mapped to an $\alpha_j$, and not some other element in $L$?
Thanks