nonarchimedean

Non-Archimedean (also written nonarchimedean) is a term used in mathematics to describe structures that fail to satisfy the Archimedean property in a meaningful way, most often in the context of valuations and metrics. A valuation v on a field K is called non-Archimedean if it satisfies the ultrametric inequality |x+y| ≤ max(|x|, |y|) for all x, y in K. The induced metric d(x,y) = |x−y| is then ultrametric. This contrasts with Archimedean valuations, such as the standard absolute value on the real or complex numbers, which only satisfy the ordinary triangle inequality.

Ultrametric spaces arising from non-Archimedean valuations have distinctive topological features. The strong triangle inequality implies that every triangle is isosceles with at least two equal longer sides, and balls exhibit unusual but well-behaved structure: open balls are also closed (clopen), and the space is typically totally disconnected. These properties influence convergence, continuity, and analytic methods in such settings.

Prominent examples include the p-adic absolute value on the rational numbers, leading to the p-adic numbers Q_p and p-adic analysis. Other examples come from fields with t-adic valuations on Laurent series F((t)) and more general non-Archimedean fields used in number theory and algebraic geometry.

Non-Archimedean concepts underpin areas such as rigid analytic geometry and Berkovich spaces, which adapt complex-analytic ideas to non-Archimedean settings. They play a central role in p-adic analysis, Diophantine geometry, and various aspects of arithmetic geometry.