nLab absolute value

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

This means it is a function

$${\vert {-} \vert}\colon k \to \mathbb{R}$$

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

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

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

3. ${\vert x \cdot y \vert} = {\vert x \vert} {\vert y \vert}$;

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

If the last triangle inequality is strengthened to

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

then ${\vert {-} \vert}$ is called an ultrametric or non-archimedean absolute value, since then for any $x, y \in k$ with $\vert x \vert \lt \vert y \vert$ then for all natural numbers $n$, $\vert n x \vert \leq \vert x \vert \lt \vert y \vert$. If the opposite holds, that whenever $\vert x \vert \lt \vert y \vert$ (and $x\neq 0$) there exists a natural number $n$ with $\vert n x \vert \gt \vert y \vert$, then it is called archimedean.

Two absolute values ${\vert {-} \vert}_1$ and ${\vert {-} \vert}_2$ are called equivalent if for all $x \in k$

$$({\vert x \vert}_1 \lt 1) \Leftrightarrow ({\vert x \vert}_2 \lt 1) \,.$$

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 ${\vert {-} \vert}_0$ defined by

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

This is non-archimedean.

On the real and complex numbers

Since the real numbers are a sequentially Cauchy complete Archimedean field, the standard absolute value ${\vert {-} \vert_\infty}$ on the real numbers is

$${\vert x \vert_\infty} = \lim_{n \to \infty} x \tanh(n x)$$

With the standard absolute value function defined, the maximum and principal square root functions could be defined on the real numbers as well.

The standard absolute value on the complex numbers is

$${\vert x + i y \vert_\infty} = \sqrt{x^2 + y^2} \,.$$

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

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

whenever $n$ is an integer and $k$ and $l$ are nonzero integers not divisible by $p$ (and ${\vert 0 \vert_{p,\epsilon}} = 0$).

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

The $p$-adic absolute value is non-archimedean. The completion $\mathbb{Q}_p$ 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 $k[ [ T] ]$ over a field $k$ is a complete field with respect to the absolute value that sends a series to $\epsilon^n$ for a fixed $0 \lt \epsilon \lt 1$ and with $n$ the lowest integer such that the $n$th coefficient of the series is not $0$.

See also

• real square root

• sign function

References

• Siegfried Bosch, Ulrich Güntzer, Reinhold Remmert, §1.5, §1.6 Non-Archimedean Analysis – A systematic approach to rigid analytic geometry, 1984 (pdf)

Discussion in point-free topology:

• Ming Ng, Steven Vickers, A Point-Free Look at Ostrowski’s Theorem and Absolute Values [arXiv:2308.14758]