Let us define the square root of the discriminant of $n$ variables over a field $F$ to be $\sqrt{\Delta} = \prod_{1\le i<j\le n} (x_i − x_j )$.
Now suppose $f ∈ F[x_1,..., x_n]$ such that $τ ·f= −f$ for every transposition $τ ∈ S_n$. I am thinking how to prove that $f = B\sqrt{ ∆}$ for some $B ∈ F[σ_1,..., σ_n]$.
My classmate asked a related question Transpositions and Symmetric polynomials though I'm not able to discuss there. Anyway my approach is to use Theorem 2.4.4 in Galois theory textbook by David Cox which says: If $f ∈ F[x_1,..., x_n]$ is invariant under the alternating group $A_n$(or the subgroup of permutations that fix $\sqrt{\Delta}$), then there are $A,B ∈ F[σ_1,..., σ_n]$ such that $f = A + B\sqrt{\Delta}$. Furthermore, $f ∈ F[x_1,..., x_n]$ implies that $A,B ∈ F[σ_1,..., σ_n]$.
But the proof of this theorem was not given and I don't understand how the statement I want to prove above follows from this theorem, so appreciate any help to make the link more precise~