# $p$-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.