Padic

Padic refers to p-adic numbers, a system in number theory associated with a prime p. It is based on the p-adic valuation and the corresponding p-adic absolute value, which yields a non-Archimedean metric. The p-adic numbers form the field Q_p, the completion of the rational numbers with respect to this metric.

Definition and expansion: For a nonzero rational x, write x = p^k · u with u a rational not divisible by p. Then v_p(x) = k and |x|_p = p^{-k}. The p-adic integers Z_p consist of elements with |x|_p ≤ 1; every element of Z_p can be written as a power series sum_{i=0}^∞ a_i p^i with digits a_i in {0, ..., p-1}. More generally, elements of Q_p have a representation sum_{i=m}^∞ a_i p^i with m in Z.

Construction and structure: Z_p is the inverse limit of the rings Z/p^nZ and is a compact, complete topological ring. Q_p is a field complete with respect to |·|_p and is locally compact. The p-adic absolute value satisfies the ultrametric inequality |x+y|_p ≤ max(|x|_p, |y|_p).

Applications: p-adic analysis and algebraic number theory use p-adic numbers to study congruences, Diophantine equations, and local-global principles. They provide local fields for primes p and underpin methods such as Hensel's lemma, as well as aspects of arithmetic geometry and Iwasawa theory.

History: p-adic numbers were introduced by Kurt Hensel in 1897 to refine lifting solutions of congruences modulo p^n to actual integer solutions. They are now a standard tool in modern number theory.