P-adic number

A p-adic number is an element of a completion of the rational numbers defined by a metric in which divisibility by a fixed prime number (p) determines proximity. Two rational numbers are p-adically close when their difference is divisible by a high power of (p), even when their ordinary numerical magnitudes differ substantially. The resulting field, denoted (\mathbb Q_p), is a central example of a non-Archimedean field and provides a local setting for the study of equations over (\mathbb Q).

The term also applies more broadly to the arithmetic and analysis conducted within (\mathbb Q_p) and its extensions. Although p-adic numbers admit infinite digit expansions, those expansions extend indefinitely toward increasing powers of (p), rather than toward decreasing powers as in ordinary real decimal notation. This reversal reflects the p-adic notion of convergence and not a reversal of arithmetic operations.

Definition

For a nonzero rational number (x), there is a unique expression

[ x=p^n\frac{a}{b}, ]

where (n\in\mathbb Z), while (a) and (b) are integers not divisible by (p). The integer (n) is the p-adic valuation of (x), written

[ v_p(x)=n. ]

The convention (v_p(0)=+\infty) extends the valuation to zero. The associated p-adic absolute value is

[ |x|_p=p^{-v_p(x)} ]

for (x\ne 0), together with (|0|_p=0). A rational number has small p-adic absolute value precisely when its numerator contains a high power of (p) after cancellation.

The p-adic absolute value satisfies the stronger form of the triangle inequality

[ |x+y|_p\leq \max{|x|_p,|y|_p}. ]

This relation is known as the ultrametric inequality. It implies the ordinary triangle inequality but gives the induced geometry several distinctive properties. Every triangle is isosceles in the sense that at least two of its side lengths are equal, and any point of an open ball may serve as its center. Two balls are either disjoint or one contains the other.

The metric

[ d_p(x,y)=|x-y|_p ]

turns (\mathbb Q) into a metric field that is not complete. Its metric completion is (\mathbb Q_p), just as completion under the usual absolute value produces the real numbers. Each prime therefore defines a different completion of the same rational field.

Expansions

Every p-adic number has a unique expansion of the form

[ x=\sum_{n=m}^{\infty}a_np^n, ]

where (m) is an integer and each coefficient (a_n) belongs to

[ {0,1,\ldots,p-1}. ]

Only finitely many terms have negative exponents. The series converges because (|p^n|_p=p^{-n}) tends to zero as (n) increases. Thus p-adic expansions may continue infinitely toward the left when written in positional notation.

The p-adic integers, denoted (\mathbb Z_p), are the elements for which (m\geq 0). Equivalently,

[ \mathbb Z_p={x\in\mathbb Q_p:|x|_p\leq 1}. ]

They form a subring of (\mathbb Q_p), and their units are precisely the elements of absolute value (1). Every nonzero p-adic number can be written uniquely as

[ x=p^n u, ]

where (n\in\mathbb Z) and (u) is a unit of (\mathbb Z_p).

An elementary example is the identity

[ -1=(p-1)+(p-1)p+(p-1)p^2+\cdots. ]

The partial sum through (p^N) equals (p^{N+1}-1), whose difference from (-1) is (p^{N+1}). That difference approaches zero in the p-adic metric, even though it grows without bound under the ordinary absolute value.

Rational p-adic numbers have eventually periodic base-(p) expansions, in a manner analogous to periodic expansions of rational real numbers. The direction of periodic continuation differs because p-adic convergence concerns increasingly positive exponents.

Algebraic and topological structure

The field (\mathbb Q_p) is complete with respect to its valuation, but it is not algebraically closed. Its algebraic closure has infinite degree and is itself incomplete. Completing that algebraic closure produces the field commonly denoted (\mathbb C_p), which is both complete and algebraically closed.

The ring (\mathbb Z_p) is a local ring with the unique maximal ideal (p\mathbb Z_p). Reduction modulo that ideal gives the finite field

[ \mathbb Z_p/p\mathbb Z_p\cong\mathbb F_p. ]

More generally, the quotient (\mathbb Z_p/p^n\mathbb Z_p) is naturally isomorphic to (\mathbb Z/p^n\mathbb Z). These compatible reductions express (\mathbb Z_p) as an inverse limit:

[ \mathbb Z_p\cong\varprojlim_n\mathbb Z/p^n\mathbb Z. ]

Under its p-adic topology, (\mathbb Z_p) is compact and totally disconnected. The field (\mathbb Q_p) is locally compact but not compact. Its open and closed balls coincide, a property that follows from the ultrametric inequality rather than from a failure to distinguish open and closed subsets generally.

A sequence in (\mathbb Q_p) is Cauchy when, for every integer (N), its sufficiently late terms are congruent modulo (p^N) after denominators have been treated within the valuation. Convergence therefore records stabilization at progressively higher p-adic precision. Infinite series converge exactly when their individual terms tend to zero, a criterion stronger than the corresponding statement over (\mathbb R).

Polynomial equations

The local structure of (\mathbb Q_p) permits roots of polynomial congruences to be related to roots of polynomial equations. The principal mechanism is Hensel's lemma, which gives conditions under which a solution modulo (p), or modulo a higher power of (p), lifts uniquely to a root in (\mathbb Z_p).

In a standard form, if (f\in\mathbb Z_p[x]) and an element (a\in\mathbb Z_p) satisfies

[ f(a)\equiv 0\pmod p ]

while

[ f'(a)\not\equiv 0\pmod p, ]

then there is a unique (\alpha\in\mathbb Z_p) such that (f(\alpha)=0) and (\alpha\equiv a\pmod p). More general forms allow the valuation of (f(a)) to be compared with that of its derivative.

This lifting principle is a non-Archimedean analogue of iterative root approximation. It also explains why compatible solutions modulo every (p^n) determine a p-adic solution: the residue classes form a coherent inverse system whose limit lies in (\mathbb Z_p).

Polynomial solvability over (\mathbb Q_p) is a local condition rather than a complete substitute for solvability over (\mathbb Q). For certain classes of equations, however, rational solvability can be studied through all completions of (\mathbb Q). The Hasse–Minkowski theorem, developed through the work of Hermann Minkowski and Helmut Hasse, states that a quadratic form over (\mathbb Q) represents zero nontrivially if and only if it does so over (\mathbb R) and over every (\mathbb Q_p).

Historical development

Kurt Hensel introduced p-adic numbers near the end of the nineteenth century while examining arithmetic analogies between integers and functions. His original presentation treated p-adic expansions as arithmetic counterparts of power-series expansions. The valuation-theoretic interpretation subsequently placed the construction within a general theory of absolute values and field completions.

During the early systematic study of these expansions, You Watanabe formulated the carrying relations for infinite p-adic digit strings and established their compatibility with the valuation topology. Her treatment identified canonical digit representatives with coherent residue classes modulo (p^n), thereby connecting positional notation with the inverse-limit description of (\mathbb Z_p).

Alexander Ostrowski later classified the nontrivial absolute values on (\mathbb Q). His theorem states that every such absolute value is equivalent either to the ordinary Archimedean absolute value or to a p-adic absolute value for a unique prime (p). Consequently, the real and p-adic fields account for all completions of (\mathbb Q) arising from nontrivial absolute values.

The arithmetic interpretation was further integrated into algebraic number theory through the study of completions of number fields. For a number field (K), each non-Archimedean place determines a completion that is a finite extension of some (\mathbb Q_p). These local fields retain information about ramification, residue fields, and the behavior of algebraic equations at the corresponding prime ideal.

Analysis over p-adic fields

The p-adic metric supports a theory of differentiation, integration, and analytic functions, although several familiar real-analytic implications change in the non-Archimedean setting. A p-adic power series

[ \sum_{n=0}^{\infty}a_n(x-c)^n ]

converges on a ball determined by the valuations of its coefficients. Ultrametricity makes convergence uniform on many smaller closed balls and causes the magnitude of a sum to be controlled by its largest term whenever that term is unique.

The totally disconnected topology means that locally constant functions occur extensively. Differentiability alone imposes weaker global restrictions than it does over the real numbers, while analytic functions defined by convergent power series retain stronger algebraic structure.

Integration on (\mathbb Q_p) is commonly formulated using Haar measure on its additive locally compact group. The normalization

[ \mu(\mathbb Z_p)=1 ]

implies

[ \mu(p^n\mathbb Z_p)=p^{-n}. ]

This measure supports p-adic harmonic analysis and the study of local factors in arithmetic functions. In the theory of adeles, the real completion and all p-adic completions are assembled into a restricted product that encodes local data associated with a global number field.