Let $L$ be a finite field extension of $F$ and let $α ∈ L$. Also, let $α_1 = α, α_2, · · · , α_r$ be the distinct elements of $L$ obtained by applying the elements of $Gal(L|F)$ to $α$. Let us consider the polynomial $h(x) = \prod^r_ {j=1} (x − α_j ) ∈ L[x]$. I wish to show that there is a positive integer $m$ such that $\prod_{σ∈Gal(L|F)}(x − σ(α)) = h^m(x)$
Not sure if it's relevant but I know from a textbook proof of "$F$ is the fixed field of $Gal (L/F)$ acting on $L$ implies $F ⊂ L$ is a normal separable extension" that this $h ∈ F[x]$ and that $h$ is irreducible over $F$, thus is the minimal polynomial of $α$ over F. The above formula for $h$ also shows that $h$ is separable and splits completely over L, but I don't know how to go about proving the desired result still hmm~
Since the action of $Gal(L/F)$ on $\{\alpha_1,\dots,\alpha_r\}$ is a group action, each of the $(x-\alpha_i)$ will appear the same number of times in the product.