p-adic absolute value

Summary

For a prime $p$, the $p$-adic absolute value of a nonzero rational number $x$ is $\abs{x}_p = p^{-v_p(x)}$, where $v_p$ is the p-adic valuation. It is a non-Archimedean absolute value on $\Q$. Completing $\Q$ for it gives the field of p-adic numbers. Together with the usual absolute value, these are all the nontrivial absolute values on $\Q$ up to equivalence.