So we have the situation that $f\in K[X_1,...,X_n]$ is anti-symmetric, which means that $\sigma (f)=\pm f$ where it is a plus if $\sigma$ is an even permutation on the $X_i$ and a minus if it is not an even permutation. Now I have to prove that there exists a $g$ which is symmetric such that $f=g\prod_{1\leq i < j\leq n}(X_i-X_j)$. This is what I thought I should do to prove the statement: If we have permutation $\sigma=(12)$ ,which sends $X_1\mapsto X_2$ and $X_2\mapsto X_1$, then $\sigma(f)=-f$. This implies that $X_1-X_2$ divides $f$ (this is the statement which I am not so sure of). Now we can do this for every permutation $\sigma=(ij)$ where $i\neq j$. Then we have that $\prod_{1\leq i < j\leq n}(X_i-X_j)$ divides $f$. It follows immediately that $g$ must be symmetric.
Is this proof valid? Thanks for looking at it.