Let $K[X_1,...,X_n]$ polynomial ring in $n$ variables and denote by $S_1, S_2,..., S_n$ the elementary symmetric polynomials:
$S_1= \sum X_i$
$ S_2 = \sum_{1 \le j < k \le n}X_j X_k$
...
$S_n= X_1 X_2 ... X_n$
The question is why is the polynomial ring $K[X_1,...,X_n]$ finite module over $K[S_1,...,S_n]$?
Guess, that clever induction on number of indeterminants does the job although I not see how I can perform the induction step. The start with $n=1$ is trivial. Assume the claim is true for $n-1$ and we want it for $n$. Any hint?
Beside nice answer using Vieta's formulas I would like to know if is also possible to go ahead inductively as follows:
it's obvious that $K[X_1,...,X_n]=K[X_2,..., X_n][S_1]$ holds. By IH $K[X_2,..., X_n]$ is finite over $K[S_2,..., S_n]$. The most important question is if tensoring preserves finiteness. If yes, then $K[X_2,..., X_n] \otimes K[S_1]=K[X_2,..., X_n][S_1]=K[X_1,X_2,..., X_n]$ is finite over $K[S_2,..., S_n]\otimes K[S_1]= K[S_1, S_2,..., S_n]$.
Is my "proof" correct?