$p$-Adic Numbers

A New Arithmetic Geometry

The real numbers arise by completing the rational numbers using the ordinary absolute value. The $p$ -adic numbers arise by completing the rational numbers using the $p$ -adic absolute value.

This produces a radically different geometry.

In the real numbers, powers of large integers grow without bound:

2^n\to\infty.

In the $2$ -adic world,

2^n\to0.

The geometry is governed not by magnitude but by divisibility.

The resulting fields

\mathbb{Q}_p

are central objects in modern number theory.

The $p$ -Adic Absolute Value

Fix a prime $p$ .

Every nonzero rational number can be written uniquely as

x=p^k\frac{a}{b},

where neither $a$ nor $b$ is divisible by $p$ .

The $p$ -adic valuation is

v_p(x)=k.

The corresponding absolute value is

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

Thus divisibility by $p$ determines size.

Examples:

|25|_5=5^{-2}=\frac1{25},

|7|_5=1.

Numbers divisible by high powers of $p$ become extremely small.

Metric and Distance

The $p$ -adic absolute value defines a metric:

d_p(x,y)=|x-y|_p.

This metric satisfies the ultrametric inequality:

d_p(x,z) \le \max(d_p(x,y),d_p(y,z)).

Triangles therefore behave differently from Euclidean geometry.

Every triangle is isosceles, and often equilateral in the metric sense.

For example, if

d_p(x,y)<d_p(y,z),

then

d_p(x,z)=d_p(y,z).

Distances are dominated by the largest scale.

Completion of $\mathbb{Q}$

A sequence

(x_n)

of rational numbers is $p$ -adically Cauchy if

|x_n-x_m|_p\to0

as $n,m\to\infty$ .

The completion of $\mathbb{Q}$ under this metric is the field

\mathbb{Q}_p.

Elements of $\mathbb{Q}_p$ are limits of $p$ -adic Cauchy sequences.

This construction parallels the construction of the real numbers from ordinary Cauchy sequences.

$p$ -Adic Expansions

Every $p$ -adic number has a series expansion

a_0+a_1p+a_2p^2+\cdots,

where

0\le a_i<p.

Unlike decimal expansions, the series extends infinitely to the left in divisibility rather than in magnitude.

For example, in the $5$ -adic numbers,

\cdots +2\cdot5^3+4\cdot5^2+1\cdot5+3

represents a valid $5$ -adic number.

The coefficients encode divisibility information.

Negative Numbers in $p$ -Adics

Negative integers acquire infinite expansions.

For example, in the $2$ -adic numbers,

-1 = 1+2+4+8+\cdots.

Indeed,

1+2+4+\cdots+2^n = 2^{n+1}-1,

and

2^{n+1}\to0

in the $2$ -adic metric.

Thus the limit equals

-1.

This phenomenon illustrates how infinite sums behave differently in $p$ -adic analysis.

The Ring of $p$ -Adic Integers

The set

\mathbb{Z}_p = \{x\in\mathbb{Q}_p:|x|_p\le1\}

is called the ring of $p$ -adic integers.

Its elements have expansions

a_0+a_1p+a_2p^2+\cdots

with no negative powers of $p$ .

The unique maximal ideal is

p\mathbb{Z}_p.

Every nonzero element can be written uniquely as

p^k u,

where $u$ is a unit in $\mathbb{Z}_p$ .

Thus $\mathbb{Z}_p$ behaves like a local version of the integers centered at the prime $p$ .

Hensel Lemma

One of the most important tools in $p$ -adic analysis is Hensel lemma.

Roughly speaking, it states that approximate polynomial solutions modulo powers of $p$ can often be lifted to genuine $p$ -adic solutions.

For example, suppose

f(a)\equiv0\pmod p

and

f'(a)\not\equiv0\pmod p.

Then there exists a unique $p$ -adic root near $a$ .

Hensel lemma is a $p$ -adic analogue of Newton iteration.

It allows local polynomial equations to be solved systematically.

Local Fields

The field

\mathbb{Q}_p

is the basic example of a local field.

A local field is a complete field with respect to a discrete valuation and finite residue field.

Local fields support rich analytic and algebraic structures:

power series methods,
harmonic analysis,
Galois theory,
representation theory.

They are the local building blocks of global arithmetic.

Solving Equations Locally

Many Diophantine problems are first studied over

\mathbb{Q}_p.

For example, one may ask whether

x^2+y^2=z^2

has nontrivial solutions over every $p$ -adic field.

If an equation has no solution in some $\mathbb{Q}_p$ , then it has no rational solution.

This principle is fundamental in arithmetic geometry.

Local-Global Principle

The interplay between rational numbers and all completions leads to the local-global philosophy.

A typical question asks:

If an equation has solutions in:

$\mathbb{R}$ ,
every $\mathbb{Q}_p$ ,

must it have a rational solution?

Sometimes the answer is yes, as in the Hasse-Minkowski theorem for quadratic forms.

Sometimes it is no.

Understanding these failures became one of the major themes of twentieth-century arithmetic geometry.

Haar Measure and Analysis

The field

\mathbb{Q}_p

supports integration and harmonic analysis.

Compact subsets behave differently from real analysis:

closed balls are compact,
open balls are closed,
spaces become totally disconnected.

Nevertheless, Fourier analysis and measure theory still exist.

These structures are essential in automorphic forms and representation theory.

Structural Importance

$p$ -adic numbers transformed number theory by introducing local analytic methods into arithmetic.

They now play central roles in:

local field theory,
Galois representations,
modular forms,
arithmetic geometry,
Iwasawa theory,
the Langlands program.

The $p$ -adic world reveals arithmetic structure invisible from the viewpoint of ordinary real analysis.