# Proof of “Lagrange Lemma”

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

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