Ostrowski's theorem

Every nontrivial absolute value on $\mathbb{Q}$ is equivalent either to the usual absolute value or to one $p$-adic absolute value.

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.

The nontrivial non-Archimedean absolute values on a number field $K$, up to equivalence, are indexed by the nonzero prime ideals $\mathfrak{p}$ of its ring of integers. The value associated with $\mathfrak{p}$ is a fixed exponential of the discrete valuation $v_{\mathfrak{p}}$.