= Newton identities in positive characteristic
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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 \b[even all positive power sums need not determine the polynomial] when its degree meets the characteristic. The relation $2e_2=e_1p_1-p_2$ becomes $0=0$ and supplies no value for $e_2$.
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