\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{absolute value}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{terminology}{Terminology}\dotfill \pageref*{terminology} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{trivial_absolute_value}{Trivial absolute value}\dotfill \pageref*{trivial_absolute_value} \linebreak
\noindent\hyperlink{OnTheRealAndComplexNumbers}{On the real and complex numbers}\dotfill \pageref*{OnTheRealAndComplexNumbers} \linebreak
\noindent\hyperlink{on_the_rational_numbers}{On the rational numbers}\dotfill \pageref*{on_the_rational_numbers} \linebreak
\noindent\hyperlink{on_laurent_power_series}{On Laurent power series}\dotfill \pageref*{on_laurent_power_series} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{terminology}{}\subsection*{{Terminology}}\label{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).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

This means it is a [[function]]

\begin{displaymath}
{\vert {-} \vert}\colon k \to \mathbb{R}
\end{displaymath}
to the [[real numbers]] such that for all $x, y \in k$

\begin{enumerate}%
\item ${\vert x \vert} \geq 0$;


\item ${\vert x \vert} = 0$ precisely if $x = 0$;


\item ${\vert x \cdot y \vert} = {\vert x \vert} {\vert y \vert}$;


\item ${\vert x + y \vert} \leq {\vert x \vert} + {\vert y \vert}$ (the [[triangle inequality]]).



\end{enumerate}
If the last triangle inequality is strengthened to

\begin{itemize}%
\item ${\vert x + y \vert} \leq max({\vert x \vert}, {\vert y \vert})$

\end{itemize}
then ${\vert {-} \vert}$ is called an [[ultrametric]] or \textbf{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 \textbf{archimedean}.

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

\begin{displaymath}
({\vert x \vert}_1 \lt 1)
   \Leftrightarrow
  ({\vert x \vert}_2 \lt 1)
  \,.
\end{displaymath}
An [[equivalence class]] of absolute values is also called a \textbf{[[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]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{trivial_absolute_value}{}\subsubsection*{{Trivial absolute value}}\label{trivial_absolute_value}

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

\begin{displaymath}
{\vert x \vert}_0 = 
  \left\{
    \itexarray{
       0 & if\; x = 0
       \\
       1 & otherwise
    }
  \right.
  \,.
\end{displaymath}
This is non-archimedean.

\hypertarget{OnTheRealAndComplexNumbers}{}\subsubsection*{{On the real and complex numbers}}\label{OnTheRealAndComplexNumbers}

The standard absolute value ${\vert {-} \vert_\infty}$ on the [[real numbers]] is

\begin{displaymath}
{\vert x \vert_\infty} = 
  \sqrt{x^2}
  = 
  \left\{
    \itexarray{
       x & if\; x \geq 0
       \\  
       - x & otherwise
    }
  \right.
  \,.
\end{displaymath}
The standard absolute value on the [[complex numbers]] is

\begin{displaymath}
{\vert x + i y \vert_\infty}
  = 
  \sqrt{x^2 + y^2}
  \,.
\end{displaymath}
These standard absolute values are archimedean, and with respect to these standard absolute values, both $\mathbb{R}$ and $\mathbb{C}$ are [[complete field|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]].

\hypertarget{on_the_rational_numbers}{}\subsubsection*{{On the rational numbers}}\label{on_the_rational_numbers}

The standard absolute value \hyperlink{OnTheRealAndComplexNumbers}{above} restricts to the standard absolute value on the [[rational numbers]]

\begin{displaymath}
{\vert {-} \vert_\infty}\colon \mathbb{Q} \to \mathbb{R}
  \,.
\end{displaymath}
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

\begin{displaymath}
\left\vert \frac{k}{l} p^n\right\vert_{p,\epsilon}
  = 
  \epsilon^n
\end{displaymath}
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 \textbf{$p$-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 \textbf{$p$-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 [[complete field|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 $\mathbb{Q}$.

\hypertarget{on_laurent_power_series}{}\subsubsection*{{On Laurent power series}}\label{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 $0$.

\hypertarget{references}{}\subsection*{{References}}\label{references}

Section 1.5, 1.6 of

\begin{itemize}%
\item [[Siegfried Bosch]], [[Ulrich Güntzer]], [[Reinhold Remmert]], \emph{[[Non-Archimedean Analysis]] -- A systematic approach to rigid analytic geometry}, 1984 (\href{http://math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf}{pdf})

\end{itemize}
[[!redirects absolute value]] [[!redirects absolute values]]

[[!redirects archimedean absolute value]] [[!redirects Archimedean absolute value]] [[!redirects archimedean absolute values]] [[!redirects Archimedean absolute values]]

[[!redirects archimedean]] [[!redirects non-archimedean]]

[[!redirects non-archimedean valuation]] [[!redirects non-archimedean valuations]]



\end{document}