p-adic Numbers

By SkytrailGroup Mathematics Editorial Team April 12, 2026

Definition

The p-adic absolute value: |pⁿ·(a/b)|ₚ=p⁻ⁿ (p∤a,b). ℚₚ is the completion of ℚ under this metric.

Examples

Example 1. What is |12|₃?

Solution. 12=3·4, so |12|₃=3⁻¹=1/3.

In Depth

The \(p\)-adic numbers \(\mathbb{Q}_p\) are a completion of \(\mathbb{Q}\) using the \(p\)-adic absolute value \(|p^k m/n|_p=p^{-k}\) (with \(p\nmid m,n\)). Unlike the real completion, \(p\)-adic numbers measure divisibility by \(p\): numbers highly divisible by \(p\) are 'small'.

Every nonzero rational has a unique \(p\)-adic expansion \(\sum_{k=v}^\infty a_k p^k\) with \(0\leq a_k

Hensel's lemma: if \(f(a)\equiv0\pmod{p}\) and \(f'(a)\not\equiv0\pmod{p}\), then \(a\) lifts uniquely to a root of \(f\) in \(\mathbb{Z}_p\). This is the \(p\)-adic analogue of Newton's method and is used to solve congruences modulo prime powers.

Ostrowski's theorem: every non-trivial absolute value on \(\mathbb{Q}\) is equivalent to either the usual absolute value or a \(p\)-adic absolute value. So the reals and the \(p\)-adic fields are the only completions of \(\mathbb{Q}\) — they capture all possible notions of 'closeness' for rationals.

\(p\)-adic numbers are central to modern number theory: the Hasse–Minkowski theorem uses all \(p\)-adic fields simultaneously; \(p\)-adic \(L\)-functions interpolate classical \(L\)-functions; and \(p\)-adic Hodge theory connects \(p\)-adic representations to differential forms.

Key Properties & Applications

The \(p\)-adic integers \(\mathbb{Z}_p\) are the completion of \(\mathbb{Z}\) under the \(p\)-adic metric. They form a compact ring containing \(\mathbb{Z}\) as a dense subring. Every \(p\)-adic integer has a unique expansion \(\sum_{k=0}^\infty a_k p^k\) with \(0\leq a_k

The \(p\)-adic exponential and logarithm converge on appropriate domains in \(\mathbb{Z}_p\), enabling \(p\)-adic analysis. The \(p\)-adic gamma function and \(p\)-adic \(L\)-functions are analytic functions on \(\mathbb{Z}_p\) that interpolate classical number-theoretic functions.

In the Langlands program, \(p\)-adic representations of Galois groups are central objects. The \(p\)-adic Langlands correspondence (proved for \(GL_2(\mathbb{Q}_p)\) by Colmez and Kisin) relates \(p\)-adic Galois representations to \(p\)-adic representations of \(GL_2(\mathbb{Q}_p)\).