$p$-Adic Numbers

Beyond the Real Numbers

The real numbers arise by completing the rational numbers with respect to the ordinary absolute value. This completion produces a field suited to Euclidean geometry and classical analysis.

Number theory reveals another completion process based on divisibility by a prime number $p$ . The resulting fields are the $p$ -adic numbers.

While real analysis measures magnitude geometrically, $p$ -adic analysis measures arithmetic proximity. Two numbers are close $p$ -adically when their difference is divisible by a high power of $p$ .

This alternative geometry lies at the center of modern arithmetic.

The $p$ -Adic Absolute Value

Fix a prime number $p$ .

Every nonzero rational number can be written uniquely in the form

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

where $a$ and $b$ are integers not divisible by $p$ .

The $p$ -adic absolute value is defined by

|x|_p=p^{-k}.

Additionally,

|0|_p=0.

The more divisible a number is by $p$ , the smaller it becomes $p$ -adically.

For example, in the $5$ -adic absolute value,

|25|_5=\frac1{25},

while

\left|\frac1{25}\right|_5=25.

Thus divisibility controls size.

The associated metric is

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

Two numbers are close if their difference contains a large power of $p$ .

Cauchy Sequences

A sequence

(x_n)

of rational numbers is $p$ -adically Cauchy if

|x_n-x_m|_p\to0

as $n,m\to\infty$ .

Equivalently, the differences become divisible by arbitrarily large powers of $p$ .

For example, consider the sequence

1,\quad 1+p,\quad 1+p+p^2,\quad 1+p+p^2+p^3,\quad\ldots

The difference between successive terms is

p^n,

whose $p$ -adic absolute value tends to zero:

|p^n|_p=p^{-n}\to0.

Hence the sequence is Cauchy in the $p$ -adic metric.

Although it diverges in the ordinary real sense, it converges $p$ -adically.

This illustrates how $p$ -adic geometry differs fundamentally from Euclidean geometry.

Construction of $\mathbb{Q}_p$

The field of $p$ -adic numbers is obtained by completing $\mathbb{Q}$ with respect to the $p$ -adic absolute value.

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

Definition.

\mathbb{Q}_p

is the completion of $\mathbb{Q}$ under the metric induced by $|\cdot|_p$ .

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

The field operations extend continuously from $\mathbb{Q}$ .

Thus $\mathbb{Q}_p$ is a complete field equipped with a non-archimedean absolute value.

$p$ -Adic Expansions

Every $p$ -adic number admits a series expansion analogous to decimal expansions.

Each element of $\mathbb{Q}_p$ can be written as

x=\sum_{n=k}^{\infty} a_n p^n,

where

a_n\in\{0,1,\ldots,p-1\}.

Unlike decimal expansions, the powers extend infinitely to the left in ordinary size but infinitely to the right in divisibility.

For example, in the $3$ -adic numbers,

\ldots222_3

represents a convergent infinite series:

2+2\cdot3+2\cdot3^2+\cdots.

Using the geometric series formula,

1+3+3^2+\cdots = \frac1{1-3} = -\frac12,

2(1+3+3^2+\cdots)=-1.

Thus

\ldots222_3=-1

in $\mathbb{Q}_3$ .

Infinite expansions therefore behave differently in $p$ -adic analysis.

The Ring of $p$ -Adic Integers

The subset

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

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

Elements of $\mathbb{Z}_p$ have expansions

a_0+a_1p+a_2p^2+\cdots

with no negative powers of $p$ .

The ring $\mathbb{Z}_p$ is compact, complete, and local. Its unique maximal ideal is

p\mathbb{Z}_p.

The quotient satisfies

\mathbb{Z}_p/p\mathbb{Z}_p \cong \mathbb{F}_p.

Thus $\mathbb{Z}_p$ may be viewed as a lift of the finite field $\mathbb{F}_p$ into characteristic zero.

Ultrametric Geometry

The $p$ -adic metric satisfies the ultrametric inequality:

|x+y|_p \le \max(|x|_p,|y|_p).

This has remarkable geometric consequences.

Every Triangle Is Isosceles

|x-z|_p>|x-y|_p,

then necessarily

|x-z|_p=|y-z|_p.

Thus every triangle has at least two equal longest sides.

Open Balls Are Closed

In Euclidean geometry, open and closed sets differ sharply. In $p$ -adic geometry, balls behave differently.

A $p$ -adic ball

B(a,r)=\{x:|x-a|_p<r\}

is simultaneously open and closed.

Nested Structure

Two $p$ -adic balls are either disjoint or one contains the other.

This produces a highly hierarchical geometry resembling a tree.

The topology of $\mathbb{Q}_p$ therefore differs radically from that of $\mathbb{R}$ .

Hensel’s Lemma

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

It allows approximate solutions modulo powers of $p$ to be lifted into genuine $p$ -adic solutions.

Roughly speaking, if a polynomial equation has a sufficiently nondegenerate solution modulo $p$ , then it has a solution in $\mathbb{Z}_p$ .

This principle resembles Newton’s method in classical analysis.

For example, consider

x^2-2.

Modulo $7$ ,

3^2=9\equiv2\pmod7.

Since the derivative

2x

is nonzero modulo $7$ at $x=3$ , Hensel’s lemma implies that $\sqrt2$ exists in $\mathbb{Q}_7$ .

Thus local solvability can often be studied through modular arithmetic.

Local Fields

The field $\mathbb{Q}_p$ is the fundamental example of a local field.

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

Finite extensions of $\mathbb{Q}_p$ play the same role locally that number fields play globally.

Much of modern arithmetic studies problems separately over:

\mathbb{R}, \qquad \mathbb{C}, \qquad \mathbb{Q}_p.

These local analyses are later assembled into global information.

$p$ -Adic Analysis

One can develop calculus over $\mathbb{Q}_p$ .

There are notions of:

convergence;
differentiation;
analytic functions;
integration;
exponential and logarithmic functions.

However, convergence behaves differently.

For example, the geometric series

1+x+x^2+\cdots

converges whenever

|x|_p<1.

Thus the series converges for all multiples of $p$ , regardless of their ordinary size.

Many classical analytic constructions therefore possess $p$ -adic analogues.

$p$ -Adic Numbers in Number Theory

The $p$ -adic numbers are indispensable in modern arithmetic.

They appear in:

local-global principles;
Diophantine equations;
Galois representations;
modular forms;
elliptic curves;
Iwasawa theory;
arithmetic geometry.

A Diophantine equation is often first studied locally in every field

\mathbb{Q}_p

and over $\mathbb{R}$ . Failure of solvability in one completion immediately prevents global rational solutions.

Thus the $p$ -adic numbers provide local windows into global arithmetic structure.

Their introduction transformed number theory from a theory of integers into a geometric and analytic theory of local fields.