Galois group and discriminant $\Delta_f^2$

Question

Let $K$ be a field with $\mathrm{char} K \neq 2$ and $f(x)\in K[x]$ a polynomial of degree $n$ with $n$ distinct roots $u_1, \ldots, u_n$ in a splitting field $F$ of $f$ over $K$. Let $\Delta_f = \prod_{i<j} (u_i-u_j)\in F$.

I want to see following two things:

$\mathrm{Gal}(F/K)$ is a subgroup of the group of all even permutations of $\{u_1,...,u_n\}$ if and only if $\Delta_f\in K$.

and

$\Delta_f^2\in K$.

Helps are much appreciated. This question is related but I don't understand the answer. Can anyone provide more details?

Answer 1

Usually $\Delta=\prod_{i\ne j}(u_i-u_j)\in F$ is for the square of $\delta=\prod_{i<j} (u_i-u_j)\in \overline{F}$ where $u_j$ are the roots of $f\in F[x]$ separable.

Given $\sigma\in \operatorname{Gal}(\overline{F}/F)$ then $u_j\to \sigma(u_j)$ is a permutation of the roots of $f$ so that $\sigma(\Delta)=\Delta$ and $\sigma(\delta) = \pm \delta$, where the sign is $+$ iff $\sigma$ is an even permutation (ie. even number of transpositions).

Then $\delta\in F$ iff $\forall \sigma\in \operatorname{Gal}(\overline{F}/F)$, $\sigma(\delta)=\delta$ iff the group of permutation obtained by letting $\operatorname{Gal}(\overline{F}/F)$ act on the roots of $f$ is a group of even permutations (ie. a subgroup of the alternating group $A_n$ where $n=\deg(f)$)

It works the same when replacing $\overline{F}$ by $K$ any normal extension of $F$ containing all the roots of $f$.