I've been trying to prove the Newton identity: $N_k(X) -s_1(X)N_k(X)+ s_2(X)N_{k-2}(X) + \dots + (-1)^{k}ks_k(X) = 0$. Where $N_k$ are the newton sums in $n$ variables and $s_k$ the elemental symetric polynomial. I've found diferent proofs using generating functions but I'm looking for an elemental proof.
I've worked up the case $n=k$ but then I'm stuck. Doing induction on k doesn't seem to be going anywhere so I was thinking in doing something in the number of variables.
