= Multiplicativity of the nonarchimedean polynomial norm
{title2=$\|fg\|_G=\|f\|_G\|g\|_G$}
For a <field> with a <Non-Archimedean absolute value>, define the coefficient norm $\|f\|_G=\max_i|a_i|$ for $f(T)=\sum_i a_iT^i$. Then $\|fg\|_G=\|f\|_G\|g\|_G$. For nonzero <polynomials>, scale each by one of its coefficients of maximum absolute value; the resulting <polynomials> have integral coefficients and at least one <unit> coefficient. Their reductions in the <residue field> are nonzero, so their product is nonzero. The product therefore also has coefficient norm $1$, proving the formula after reversing the scaling. This compares binary-polynomial coefficient heights with the heights of their roots.
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