---
date: 2022-01-15 21:49
modification date: Monday 24th January 2022 11:46:11
title: absolute value
aliases: ["absolute value", "archimedean", "nonarchimedean", "ultrametric", "ultrametric triangle inequality"]

---

Tags: 
#NT/algebraic
Refs:
[valuation](Unsorted/Valuations.md)

# Overview of Fields

- Archimedean fields: $\QQ, \RR, \CC$.
- Nonarchimedean fields: everything else.
- Global fields: 
	- Finite extension of $\QQ$ or $\FF_p(t)$, or the function field of a [geometrically integral](geometrically%20integral) curve over $\FF_{p^n}$
	  - Equivalently: $\ff(A)$ for $A\in \Alg\slice{\ZZ}^{\fg}$ with $A$ an integral domain and $\krulldim(A) = 1$.
- Local fields:
  - Finite extension of $\RR, \QQ, \QQpadic, \FF_p((t))$.
  - $\RR, \CC, \FF_{p^n}((t))$, or a finite extension of $\QQpadic$.
  - $\RR, \CC, \ff(R)$ for $R$ a complete DVR with finite residue field
  - $k$ a nondiscrete locally compact Hausdorff topological ring.
  - $k$ the completion of a global field with respect to a nontrivial absolute value.

# Archimedean and nonarchimedean 
Idea: an ordered group $K$ is **archimedean** if it does not admit any infinitesimals or infinites: pairs $(x, y)$ such that $nx < y$ for all $n$, resp. $y$ such that $n1 < y$ for all $n$.

In the presence of an absolute value, $K$ is archimedean iff
$$x\in K \implies \exists n\text{ such that } \abs{nx} > 1$$
# Absolute values

An **absolute value** on a field $k$ is a map $\abs{\wait}: k \rightarrow \mathbb{R}_{\geq 0}$ such that for all $x, y \in k$ the following hold:

- $|x|=0$ if and only if $x=0$;
- $|x y|=|x||y|$;
- $|x+y| \leq|x|+|y|$.

If in addition the stronger **ultrametric triangle inequality** holds,
$$|x+y| \leq \max (|x|,|y|),$$
then the absolute value is **nonarchimedean**; otherwise it is **archimedean**.

Two absolute values $\abs{\wait}_1$ and $\abs{\wait}_2$ on $K$ are **equivalent** if there exists an $\alpha \in \mathbb{R}_{>0}$ for which $|x|_2=|x|_1^{\alpha}$ for all $x \in k$.

# Results
- Every absolute value induces a metric $d(x, y) \da \abs{x-y}$, but not every metric is induced this way.
- Absolute values only exist for integral domains, and extend to fraction fields.
- If $\characteristic K > 0$, every absolute value is nonarchimedean.
- If $\FF_q$ is a finite field, there is only a trivial absolute value.
- Get an absolute value from a [valuation](Unsorted/Valuations.md): $$\abs{x}_p \da p^{-v_p(x)},\qquad \abs{0}_p \da p^{-\infty} = 0$$
- If an integral domain is [Cauchy complete](Unsorted/Cauchy%20completion.md) with respect to an absolute value it is Cauchy complete with respect to all equivalent absolute values.

## Topological Results
- [Weak approximation](Weak%20approximation.md) implies that two absolute values on the same field induce the same topology if and only if they are equivalent.
- ![](attachments/Pasted%20image%2020220124115854.png)
- Translation is a homeomorphism.
Thus in order to understand the topology of a topological group, we can focus on neighborhoods of the identity; a base of open neighborhoods about the identity determines the entire topology.
![](attachments/Pasted%20image%2020220124120329.png)
![](attachments/Pasted%20image%2020220124225131.png)

# Examples

- $\CC,\RR,\QQ$ with the Euclidean absolute value are archimedean.
- Rational function fields $K\functionfield{x}$ and formal Laurent series fields $K\fls{x}$ are nonarchimedean.
- $\ZZpadic$ and finite extensions of $\QQpadic$ are nonarchimedean.
- The map $\abs{\wait}: k \rightarrow \mathbb{R}_{\geq 0}$ defined by
$$
|x|= \begin{cases}1 & \text { if } x \neq 0, \\ 0 & \text { if } x=0,\end{cases}
$$
	is the trivial absolute value on $k$. It is nonarchimedean.

- [p-adic](Unsorted/p-adic.md) as an [adic completion](Unsorted/adic%20completion.md) and a [Cauchy completion](Unsorted/Cauchy%20completion.md)
	![](attachments/Pasted%20image%2020220124120916.png)
- [valuation ring](Unsorted/Valuations.md) of the [adic completion](Unsorted/adic%20completion.md) of a rational [Unsorted/function field](Unsorted/function%20field.md) over $K = \FF_q$:
![](attachments/Pasted%20image%2020220124121057.png)