nLab
absolute value

Contents

Terminology

In field theory, what we call an ‘absolute value’ here is often called a ‘valuation’. However, there is also a more general notion of valuation used in field theory, which is what we call ‘valuation’. The notion of absolute value is also used in functional analysis, where it may be called a ‘multiplicative norm’ (rather than merely submultiplicative, as norms on Banach algebras are required to be).

Definition

For kk a rig (typically either a field or at least an integral domain, or else an associative algebra over such), an absolute value on kk is a (non-trivial) multiplicative seminorm, or equivalently a finite real-valued valuation.

This means it is a function

||:k {\vert - \vert}\colon k \to \mathbb{R}

to the real numbers such that for all x,ykx, y \in k

  1. |x|0{\vert x \vert} \geq 0;

  2. |x|=0{\vert x \vert} = 0 precisely if x=0x = 0;

  3. |xy|=|x||y|{\vert x \cdot y\vert} = {\vert x \vert} {\vert y \vert};

  4. |x+y||x|+|y|{\vert x + y\vert} \leq {\vert x \vert} + {\vert y \vert} (the triangle inequality).

If the last trinagle inequality is strengthened to

  • |x+y|max(|x|,|y|){\vert x + y\vert} \leq max({\vert x \vert}, {\vert y \vert})

then ||{\vert -\vert} is called an ultrametric or non-archimedean absolute value. Otherwise it is called archimedean.

Two absolute values || 1{\vert - \vert}_1 and || 2{\vert - \vert}_2 are called equivalent if for all xkx \in k

(|x| 1<1)(|x| 2<1). ({\vert x \vert}_1 \lt 1) \Leftrightarrow ({\vert x \vert}_2 \lt 1) \,.

An equivalence class of absolute values is also called a place.

A field equipped with an absolute value which is a complete metric space with respect to the corresponding metric is called a complete field.

Examples

Trivial absolute value

Every field admits the trivial absolute value || 0{\vert- \vert}_0 defined by

|x| 0={0 ifx=0 1 otherwise. {\vert x \vert}_0 = \left\{ \array{ 0 & if\; x = 0 \\ 1 & otherwise } \right. \,.

This is non-archimedean.

On the real and complex numbers

The standard absolute value || {\vert -\vert}_\infty on the real numbers is

|x| =x 2={x ifx0 x otherwise. {\vert x \vert}_{\infty} = \sqrt{x^2} = \left\{ \array{ x & if x \geq 0 \\ - x & otherwise } \right. \,.

The standard absolute value on the complex numbers is

|x+iy|=x 2+y 2. {\vert x + i y \vert} = \sqrt{x^2 + y^2} \,.

These standard absolute values are archimedean, and with respect to these standard absolute values, both \mathbb{R} and \mathbb{C} are complete and hence are complete archimedean valued fields. Notice that \mathbb{R} is in addition an ordered field and as such also an archimedean field.

On the rational numbers

The standard absolute value above restricts to the standard absolute value on the rational numbers

|| :. {\vert -\vert}_\infty\colon \mathbb{Q} \to \mathbb{R} \,.

Moreover, for any prime number pp there is a (class of an) absolute value || p{\vert -\vert}_p on \mathbb{Q} defined by

|klp n| p=ϵ n {\vert \frac{k}{l} p^n\vert}_p = \epsilon^n

for any 0<ϵ<10 \lt \epsilon \lt 1 and where klp n\frac{k}{l}p^n is the unique such decomposition of a given rational number with k,lmatrhbbZk, l \in \matrhbb{Z} not divisible by pp.

This is called the pp-adic absolute value. It is non-archimedean. The completion p\mathbb{Q}_p of \mathbb{Q} by this is called the field of p-adic numbers, which is therefore a non-archimedean field.

Ostrowski's theorem says that this exhaust the non-trivial absolute values on the rational numbers. Therefore the real numbers and the p-adic numbers are the only possible completions of \mathbb{Q}.

On Laurent power series

The field of Laurent series k[[T]]k[ [ T] ] over a field kk is a complete field with respect to the absolute value that sends a series to ϵ n\epsilon^n for a fixed 0<ϵ<10 \lt \epsilon \lt 1 and with nn the lowest integer such that the nnth coefficient of the series is not 00.

References

Section 1.5, 1.6 of

Revised on September 6, 2014 13:26:37 by Maciej Pirog? (87.206.118.112)