Assume $F$ is a field of zero characteristic. Denote the elementary symmetric polynomials of $n$ variables by $e_k$, $\quad k=\overline{1,n}$. Let the symbol $\sum ax_1^{i_1}\dots x_n^{i_n}$ denote the sum of all different monomials which can be obtained from $ax_1^{i_1}\dots x_n^{i_n}$ by permuting its variables. For instance, if $n=3$,
$$\sum x_1^2x_2=x_1^2x_2+x_1^2x_3+x_2^2x_1+x_2^2x_3+x_3^2x_1+x_3^2x_2$$
The polynomials $S_k=\sum_{i=1}^n x_i^k$ are called power sum symmetric polynomials. Obviously $S_1=e_1$ and $S_2=e_1^2-2e_2$. In order to prove Newton's identities one can see that for $2<k<n+1$
$$e_iS_{k-i}=\sum x_1^{k-i+1}x_2\dots x_i+\sum x_1^{k-i}x_2\dots x_ix_{i+1}$$
How can we see the above identity?