Show that the distinct elements of $\{ \sigma (\alpha) \mid \sigma \in Gal(L/F) \}$ occur in equal numbers.

Question

Exercise from David Cox's Galois Theory.

Let $L$ be a finite extension of $F$. Let $\alpha \in L$. Let $\alpha_1 = \alpha, \alpha_2, \alpha_3,\dots , \alpha_r$ be the distinct elements of $L$ obtained by applying the elements of $Gal(L/F)$ to $\alpha$. Then we define $$ h(x) = \prod_{i=1}^r (x - \alpha_i) \in L[x] $$

Show there is an integer $m$ such that $$ \prod_{\sigma \in Gal(L/F)} (x - \sigma(\alpha)) = h(x)^m $$

So basically I have to show that if $\sigma_1, \dots, \sigma_k$ map $\alpha$ to $\alpha_i$, and $\pi_1, \dots, \pi_r$ map $\alpha$ to $\alpha_j$, then $k=r$. (And then I take $m = k$.)

But then I am stuck here. How to construct bijection from the $\sigma_i$ to $\pi_i$?

Answer 1

Well, this problem is part of a more general property.

For this, consider a group homomorphism $\phi:G\rightarrow H$ between finite groups with kernel $K$. Then each image value $h\in H$ is taken on equally often, namely $|K|$ times.

For this, consider the associated surjective homomorphism $\psi:G\rightarrow G/K: g\mapsto \phi(g)K$ from which you can easily infer this result.