p-adic Numbers — Mathematics Knowledge System

Overview

The p-adic numbers ℚₚ are a completion of the rational numbers ℚ using the p-adic absolute value |·|ₚ, where a rational number is 'small' if it is divisible by a high power of the prime p. By Ostrowski's theorem, the only completions of ℚ are ℝ (using the usual absolute value) and ℚₚ for each prime p. The p-adic numbers form a locally compact field of characteristic 0 with unusual properties: every triangle is isosceles, every open ball is also closed, and ℚₚ contains roots of unity. They are indispensable in modern number theory, particularly in the proofs of Fermat's Last Theorem and the Weil conjectures.

Intuition

In the usual metric, a number is 'large' if its absolute value is large. In the p-adic metric, a number is 'small' if it is divisible by a high power of p: for instance, in ℚ₅, the number 625 = 5⁴ is very small (|625|₅ = 5^{-4}). This non-Archimedean metric makes the integers ℤ dense in ℤₚ = {x ∈ ℚₚ : |x|ₚ ≤ 1}, the p-adic integers. A p-adic number can be thought of as an 'infinite power series in p': x = a₀ + a₁p + a₂p² + … with 0 ≤ aᵢ ≤ p-1.

Formal Definition

Definition

Fix a prime p. The p-adic valuation of a nonzero rational number n/m (in lowest terms) is vₚ(n/m) = vₚ(n) - vₚ(m) where vₚ(n) is the exponent of p in the factorisation of n.

v_p\left(\frac{n}{m}\right) = v_p(n) - v_p(m), \quad v_p(0) = +\infty

p-adic valuation on ℚ

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

p-adic absolute value

|x + y|_p \leq \max(|x|_p, |y|_p)

Ultrametric (non-Archimedean) inequality

\mathbb{Q}_p = \text{completion of } \mathbb{Q} \text{ with respect to } |\cdot|_p

The p-adic numbers as a completion

\mathbb{Z}_p = \{x \in \mathbb{Q}_p : |x|_p \leq 1\} = \varprojlim \mathbb{Z}/p^n\mathbb{Z}

p-adic integers as the unit ball and as a projective limit

Notation

Notation	Meaning
$v_p(x)$	p-adic valuation of x ∈ ℚ
$\|x\|_p = p^{-v_p(x)}$	p-adic absolute value
$\mathbb{Q}_p$	Field of p-adic numbers
$\mathbb{Z}_p$	Ring of p-adic integers (unit ball in ℚₚ)
$\mathbb{F}_p = \mathbb{Z}_p / p\mathbb{Z}_p$	Residue field (finite field with p elements)

Properties

Ostrowski's theorem

\text{Every nontrivial absolute value on } \mathbb{Q} \text{ is equivalent to } |\cdot| \text{ or } |\cdot|_p \text{ for some prime } p

Hensel's lemma

\text{If } f \in \mathbb{Z}_p[x],\; f(a_0) \equiv 0 \pmod{p},\; f'(a_0) \not\equiv 0 \pmod{p},\; \text{then } \exists\, a \in \mathbb{Z}_p: f(a) = 0

Every p-adic integer has a unique base-p expansion

x \in \mathbb{Z}_p \iff x = \sum_{k=0}^{\infty} a_k p^k, \quad a_k \in \{0, 1, \ldots, p-1\}

Worked Examples

1
Factor 60 = 2² · 3 · 5.
$60 = 2^2 \cdot 3^1 \cdot 5^1$
2
The 2-adic valuation is v₂(60) = 2, so |60|₂ = 2^{-2} = 1/4.
$|60|_2 = 2^{-2} = \tfrac{1}{4}$
3
v₃(60) = 1, so |60|₃ = 3^{-1} = 1/3.
$|60|_3 = 3^{-1} = \tfrac{1}{3}$
4
v₅(60) = 1, so |60|₅ = 5^{-1} = 1/5.
$|60|_5 = 5^{-1} = \tfrac{1}{5}$

✓ Answer

|60|₂ = 1/4, |60|₃ = 1/3, |60|₅ = 1/5. (Verify: 1/4 · 1/3 · 1/5 · 60 = 1 — the product formula.)

Practice Problems

Hardfree response

Prove that in the p-adic metric, every open ball B(a, r) is also closed.

Hardfree response

State the product formula for absolute values on ℚ and verify it for x = 12.

Common Mistakes

Common Mistake

Thinking ℤₚ is the same as ℤ/pℤ

ℤₚ is the ring of p-adic integers — a massive ring containing all infinite base-p expansions. It is the projective limit of ℤ/p^nℤ and surjects onto ℤ/pℤ = 𝔽ₚ as its residue field.

Common Mistake

Assuming the p-adic numbers are ordered like ℝ

ℚₚ has no natural ordering compatible with its field structure. Unlike ℝ, ℚₚ is not an ordered field. Moreover, -1 can be a square (e.g. in ℚ₅).

Quiz

In the p-adic metric, the number p^n is:

Very large (|p^n|_p = p^n)Very small (|p^n|_p = p^{-n} → 0 as n → ∞)Equal to 1 in absolute valueUndefined

Hensel's lemma allows you to:

Factor polynomials over ℤₚ into linear factorsLift a simple root of f mod p to an exact root in ℤₚShow ℤₚ is algebraically closedFind all zeros of f in ℝ from p-adic information

Historical Background

Kurt Hensel introduced p-adic numbers in 1897, motivated by the analogy between polynomials over a field and integers: just as formal power series extend polynomials, p-adic numbers extend integers. Hensel's lemma — a p-adic lifting theorem — allows solutions mod p to be lifted to p-adic solutions. Ostrowski's 1916 theorem classified all absolute values on ℚ. The 20th century saw p-adic numbers become central to algebraic number theory and arithmetic geometry.

1897
Hensel introduces p-adic numbers via Hensel's lemma
Kurt Hensel
1916
Ostrowski classifies all absolute values on ℚ
Alexander Ostrowski
1961
Tate's thesis uses p-adic analysis to prove the functional equation of L-functions
John Tate
1995
Wiles uses p-adic deformations of Galois representations in proving Fermat's Last Theorem
Andrew Wiles

Summary

The p-adic absolute value |x|_p = p^{-vₚ(x)} measures divisibility by p; ℚₚ is the completion of ℚ under this metric.
ℚₚ is a non-Archimedean locally compact field: |x+y|_p ≤ max(|x|_p, |y|_p).
By Ostrowski's theorem, ℝ and ℚₚ (for primes p) are the only completions of ℚ.
Hensel's lemma allows lifting roots mod p to roots in ℤₚ, an analogue of Newton's method.
p-adic methods are central to modern number theory: Tate's thesis, Fermat's Last Theorem, p-adic L-functions.

References

BookGouvêa, F.Q. — p-adic Numbers: An Introduction, 2nd ed. (1997), Springer
BookNeukirch, J. — Algebraic Number Theory (1999), Springer, Chapter II
BookKoblitz, N. — p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd ed. (1984), Springer
WebsiteWikipedia — p-adic number
WebsiteMathWorld — p-adic Number
WebsitenLab — p-adic number