nLab
absolute value

Context

Analysis

Algebra

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 triangle 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, since then for any x,ykx, y \in k with |x|<|y|\vert x \vert \lt \vert y \vert then for all natural numbers nn, |nx||x|<|y|\vert n x \vert \leq \vert x \vert \lt \vert y \vert. If the opposite holds, that whenever |x|<|y|\vert x \vert \lt \vert y \vert (and x0x\neq 0) there exists a natural number nn with |nx|>|y|\vert n x \vert \gt \vert y \vert, then 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_\infty} = \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.

Similar norms exist on the quaternions and octonions, showing that absolute values can be of interest on noncommutative and even nonassociative division rings.

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 and positive number ϵ<1\epsilon \lt 1, there is an absolute value || p,ϵ{\vert {-} \vert_{p,\epsilon}} on \mathbb{Q} defined by

|klp n| p,ϵ=ϵ n \left\vert \frac{k}{l} p^n\right\vert_{p,\epsilon} = \epsilon^n

whenever nn is an integer and kk and ll are nonzero integers not divisible by pp (and |0| p,ϵ=0{\vert 0 \vert_{p,\epsilon}} = 0).

These are called the pp-adic absolute values. Given pp, they are all equivalent (the open unit ball consists of all rational numbers whose denominator in lowest terms is not divisible by pp), so there is a unique pp-adic place. For most purposes, only the place matters, and one may write simply |q| p|q|_p; however, if one wants a specific absolute value, then the usual choice is to use ϵ=1/p\epsilon = 1/p (so that |p n| p=p 1/n{|p^n|_p} = p^{1/n} whenever nn is an integer).

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

Ostrowski's theorem says that these examples exhaust the non-trivial absolute values on the rational numbers. Therefore the real numbers and the p-adic numbers are the only possible field 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 April 18, 2017 12:09:03 by Mike Shulman (76.167.222.204)