Newton identities in positive characteristic

The polynomial identities remain true in positive characteristic of a field, but recovering elementary symmetric polynomials by division can fail. In the prime field ${\mathbb{F}}_2$ of characteristic two, the root lists $(0,0)$ and $(1,1)$ have the same power sum zero for every positive exponent: $0^k+0^k=0$ and $1^k+1^k=0$. Their monic polynomials differ, however: $z^2$ and $(z-1{)}^2=z^2+1$. Thus even all positive power sums need not determine the polynomial when its degree meets the characteristic. The relation $2e_2=e_1{p}_1-p_2$ becomes $0=0$ and supplies no value for $e_2$.