Can anyone share a link of proof of the following fact ?
Let $f(x) \in K[x] $ be an irreducible polynomial with $n$ distinct roots $r_i$ and let $g(x_1,\dots, x_n)$ and $h(x_1,\dots, x_n)$ be polynomials in $K[x_1,x_2,\dots, x_n] $ such that any permutation $\pi$ of $\{1,2,\dots, n\}$ for which $g(r_1, r_2,...,r_n)$ is different from $g(r_{\pi(1)} , r_{\pi(2)} ,...,r_{\pi(n)})$ we have $h(r_1, r_2,...,r_n)$ is different from $h(r_{\pi(1)} , r_{\pi(2)} ,...,r_{\pi(n)})$. Then $g(r_1, r_2,...,r_n) \in K(h(r_1, r_2,...,r_n))$.