Non-Archimedean absolute value

Summary

A non-Archimedean absolute value on a field is an absolute value that satisfies the strong triangle inequality $\abs{x + y} \le \max(\abs{x}, \abs{y})$, see the definition. It generalizes the usual absolute value of real numbers in a different direction: small elements stay small under addition. The p-adic absolute value is the main example, and the induced metric is an ultrametric. These absolute values are the same as real-valued valuations, up to a choice of base for the exponent.