= Integer criterion for a non-Archimedean absolute value
An <absolute value on a field> is non-Archimedean if and only if $|n\cdot1|\le1$ for every positive integer $n$. Necessity follows from the <ultrametric inequality>. Conversely the binomial theorem gives $|x+y|^m\le(m+1)\max(|x|,|y|)^m$, since every binomial coefficient has absolute value at most one. Take $m$th roots and let $m\to\infty$. In particular an <extension of an absolute value> that is non-Archimedean stays non-Archimedean.
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