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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 $g(r_1, r_2,...,r_n) \in K(h(r_1, r_2,...,r_n))$.

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  • $\begingroup$ Is $f$ irreducible in $K$, or do its roots lie in $K$ ? One thing usually contradicts the other. $\endgroup$ – darij grinberg Feb 7 at 17:05
  • $\begingroup$ @darijgrinberg I mean (f irreducible in the field K) with (r_i distnict roots). Sorry for the confusion. $\endgroup$ – cdt Feb 8 at 4:40
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    $\begingroup$ Do you know about Galois theory ? $\endgroup$ – Max Mar 28 at 22:28

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