A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric." Any nonzero rational number x can be represented by x = (p^a r)/s, where p is a prime number, r and s are integers not divisible by p, and a is a unique integer. Then define the p-adic norm of x by left bracketing bar x right bracketing bar _p = p^(-a).

Ax-Kochen isomorphism theorem | Diophantine equation | greatest dividing exponent | harmonic number | Hasse principle | local field | Mahler-Lech theorem | p-adic integer | p-adic norm | product formula | valuation | valuation theory | von Staudt-Clausen theorem