P-adic analysis

Summary

$p$-adic analysis is the analysis of functions of $p$-adic numbers, where nearness is measured by the $p$-adic absolute value instead of the usual one. This absolute value is ultrametric, which makes many things simpler than in real analysis: a series converges as soon as its terms tend to $0$. Other things behave differently: the $p$-adic integers form a compact, totally disconnected space, and the exponential series converges only on a small disc. The subject is a main tool of modern number theory, for example in $p$-adic $L$-functions and in the study of congruences.