= Non-Archimedean part of Ostrowski theorem
{title2=$|x|=p^{-c v_p(x)},\quad c>0$}
For a nontrivial <Non-Archimedean absolute value> on $\mathbb Q$, every integer has absolute value at most one. Some prime $p$ must have value less than one, or the value would be trivial on every rational number. Two distinct such primes are impossible by a <Bezout identity>. Prime factorization then gives the displayed formula. Distinct primes give inequivalent topologies, since powers of $p$ tend to zero only for the $p$-adic absolute value among these values.
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