# Valuations and Absolute Values

## Measuring Arithmetic Size

In ordinary analysis, the absolute value

$$
|x|
$$

measures the size of a real number.

In number theory, one studies more general notions of size that reflect arithmetic divisibility rather than geometric distance.

These generalized size functions are called valuations or absolute values.

They allow arithmetic to be studied analytically and lead naturally to $p$-adic numbers, local fields, and modern arithmetic geometry.

## Ordinary Absolute Value

The standard absolute value on $\mathbb{Q}$ satisfies:

1. positivity:

$$
|x|\ge0,
\qquad
|x|=0\iff x=0,
$$

2. multiplicativity:

$$
|xy|=|x||y|,
$$

3. triangle inequality:

$$
|x+y|\le |x|+|y|.
$$

This is called the Archimedean absolute value because repeated addition eventually exceeds any bound.

For example,

$$
|1000|>|1|.
$$

The usual geometry of the real line arises from this absolute value.

## $p$-Adic Valuation

Fix a prime number $p$.

Every nonzero rational number $x$ can be written uniquely as

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

where

$$
a,b
$$

are integers not divisible by $p$.

The exponent $k$ is called the $p$-adic valuation of $x$, denoted

$$
v_p(x)=k.
$$

For example,

$$
v_2(24)=3,
$$

since

$$
24=2^3\cdot3.
$$

Similarly,

$$
v_3\left(\frac{45}{14}\right)=2,
$$

because

$$
\frac{45}{14}=3^2\cdot\frac{5}{14}.
$$

Thus the valuation measures divisibility by $p$.

## $p$-Adic Absolute Value

From the valuation one defines the $p$-adic absolute value:

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

By convention,

$$
|0|_p=0.
$$

Thus numbers highly divisible by $p$ become very small.

For example,

$$
|8|_2=2^{-3}=\frac18,
$$

while

$$
|7|_2=1.
$$

This behavior is opposite to the ordinary absolute value.

Large powers of $p$ are tiny in the $p$-adic world.

## Non-Archimedean Triangle Inequality

The $p$-adic absolute value satisfies a much stronger triangle inequality:

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

$$
|x+y|_p\le\max(|x|_p,|y|_p)
$$

This is called the ultrametric inequality.

Consequences include:

- every triangle is isosceles,
- balls are simultaneously open and closed,
- nested divisibility dominates geometry.

For example, if

$$
|x|_p\ne|y|_p,
$$

then

$$
|x+y|_p
=
\max(|x|_p,|y|_p).
$$

Cancellation behaves very differently from ordinary real analysis.

## Completions

The rational numbers are incomplete with respect to the ordinary absolute value.

Completing them produces the real numbers:

$$
\mathbb{Q}\to\mathbb{R}.
$$

Similarly, completing $\mathbb{Q}$ with respect to the $p$-adic absolute value produces the field of $p$-adic numbers:

$$
\mathbb{Q}_p.
$$

Thus each prime $p$ generates its own arithmetic universe.

The fields

$$
\mathbb{Q}_p
$$

play a role analogous to the real numbers, but with arithmetic geometry governed by divisibility rather than Euclidean distance.

## Ostrowski Theorem

A remarkable theorem classifies all absolute values on $\mathbb{Q}$.

**Theorem (Ostrowski).** Every nontrivial absolute value on $\mathbb{Q}$ is equivalent either to:

- the ordinary absolute value,
- or a $p$-adic absolute value for some prime $p$.

Thus the arithmetic geometry of $\mathbb{Q}$ is completely controlled by:

- the infinite place $\infty$,
- and the finite primes $p$.

This theorem explains why primes and completions occupy such a central role in number theory.

## Valuation Rings

Given a valuation $v$, one defines the valuation ring

$$
\mathcal O_v
=
\{x:v(x)\ge0\}.
$$

For the $p$-adic valuation, this ring is

$$
\mathbb{Z}_p,
$$

the ring of $p$-adic integers.

Its maximal ideal is

$$
p\mathbb{Z}_p.
$$

Every nonzero element can be written uniquely as

$$
p^k u,
$$

where $u$ is a unit.

This decomposition mirrors prime factorization locally.

## Local-Global Philosophy

Valuations allow arithmetic problems to be studied locally.

Instead of solving equations directly over $\mathbb{Q}$, one studies them over:

$$
\mathbb{R},
\qquad
\mathbb{Q}_2,
\qquad
\mathbb{Q}_3,
\qquad
\mathbb{Q}_5,
\dots
$$

If an equation has no solution locally somewhere, then it cannot have a global rational solution.

This principle is central in Diophantine geometry.

The challenge is understanding when local solvability implies global solvability.

## Product Formula

Absolute values on $\mathbb{Q}$ satisfy a remarkable balancing law.

For every nonzero rational number $x$,

$$
|x|_\infty
\prod_p |x|_p
=
1,
$$

where the product runs over all primes.

$$
|x|_\infty\prod_p|x|_p=1
$$

This formula expresses a deep equilibrium among all places of $\mathbb{Q}$.

The infinite absolute value and all $p$-adic absolute values together completely describe arithmetic size.

## Geometric Interpretation

Valuations measure how functions vanish or blow up.

In algebraic geometry, valuations correspond to points, divisors, and local geometric behavior.

A valuation detects local arithmetic structure near a prime or near infinity.

This interpretation becomes fundamental in:

- algebraic geometry,
- arithmetic surfaces,
- rigid analytic geometry,
- Berkovich spaces.

## Structural Importance

Valuations and absolute values provide the local language of number theory.

They underlie:

- $p$-adic analysis,
- local fields,
- Hensel lemma,
- ramification theory,
- adeles and ideles,
- modern arithmetic geometry.

The transition from global arithmetic to local valuations is one of the foundational ideas of twentieth-century number theory.