= Classification of nontrivial valuations on the rational numbers
{title2=$v=c\,v_p$}
An additive nontrivial <valuation> on the rational field is equivalent to exactly one <P-adic valuation>. Integers have nonnegative values; some prime must have positive value, and <Bézout's identity> prevents two distinct primes from doing so. Unique factorization then gives the formula, with $c>0$ in the real-valued case. The trivial <valuation> is a separate possibility if admitted by the definition.
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