$p$-Adic 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...

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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.