# $p$-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,

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

so

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

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

If

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

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.