\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*{p-adic integer}

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

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

[[!include higher algebra - contents]]

\hypertarget{formal_geometry}{}\paragraph*{{Formal geometry}}\label{formal_geometry}

[[!include formal geometry -- contents]]

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

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

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{topology}{Topology}\dotfill \pageref*{topology} \linebreak
\noindent\hyperlink{relation_to_profinite_completion_of_the_integers}{Relation to profinite completion of the integers}\dotfill \pageref*{relation_to_profinite_completion_of_the_integers} \linebreak
\noindent\hyperlink{pontrajgin_duality_to_prfer_group}{Pontrajgin duality to Pr\"u{}fer $p$-group}\dotfill \pageref*{pontrajgin_duality_to_prfer_group} \linebreak
\noindent\hyperlink{AsFormalNeighbourhoodOfPrime}{As the formal neighbourhood of a prime}\dotfill \pageref*{AsFormalNeighbourhoodOfPrime} \linebreak
\noindent\hyperlink{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

For each [[prime number]] $p$ the [[ring]] of \emph{$p$-adic integers} $\mathbb{Z}_p$ is the [[formal completion]] of the ring $\mathbb{Z}$ at the [[prime ideal]] $(p)$. Geometrically this means that $\mathbb{Z}_p$ is the [[ring of functions]] on a [[formal neighbourhood]] of $p$ inside [[Spec(Z)]] (this is discussed in more detail \hyperlink{AsFormalNeighbourhoodOfPrime}{below}). Algebraically it means that the elements in $\mathbb{Z}_p$ look like [[formal power series]] where the formal variable is the prime number $p$.

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

For any [[prime number]] $p$, the [[ring]] of $p$-\textbf{adic integers} $\mathbb{Z}_p$ (which, to avoid possible confusion with the ring $\mathbb{Z}/(p)$ used in [[modular arithmetic]], is also written as $\widehat{\mathbb{Z}}_p$) may be described in one of several ways:

\begin{enumerate}%
\item To the person on the street, it may be described as (the ring of) numbers written in base $p$, but allowing infinite expansions to the left. Thus, numbers of the form

\begin{displaymath}
\sum_{n \geq 0} a_n p^n
\end{displaymath}
where $0 \leq a_n \lt p$, added and multiplied with the usual method of carrying familiar from adding and multiplying ordinary integers.


\item More abstractly, it is the [[limit]] $\underset{\leftarrow}{\lim} \mathbb{Z}/(p^n)$, in the [[category]] of (unital) [[ring|rings]], of the [[diagram]]

\begin{displaymath}
\ldots \to \mathbb{Z}/(p^{n+1}) \to \mathbb{Z}/(p^n) \to \ldots \to \mathbb{Z}/(p) .
\end{displaymath}
This is also a limit in the category of [[topological rings]], taking the rings in the diagram to have [[discrete topologies]].


\item Alternatively, it is the [[metric completion]] of the ring of [[integers]] $\mathbb{Z}$ with respect to the $p$-adic [[absolute value]]. Since addition and multiplication of integers are [[uniform space|uniformly continuous]] with respect to the $p$-adic [[absolute value]], they extend uniquely to a uniformly continuous addition and multiplication on $\mathbb{Z}_p$. Thus $\mathbb{Z}_p$ is a [[topological ring]].


\item Also $\mathbb{Z}[ [ x ] ]/(x-p)\mathbb{Z}[ [ x ] ]$, see at \emph{[[analytic completion]]}.



\end{enumerate}
Hence one also speaks of the \emph{$p$-[[adic completion]]} of the integers. See [[completion of a ring]] (which generalizes 2\&3).

There is also this characterization:

\begin{lemma}
\label{pAdicIntegersAspExtensionofFpByThemselves}\hypertarget{pAdicIntegersAspExtensionofFpByThemselves}{}
There is a [[short exact sequence]]

\begin{displaymath}
0 
    \to
  \mathbb{Z}_{p}
    \overset{p \cdot (-)}{\longrightarrow}
  \mathbb{Z}_{p}
    \longrightarrow
  \mathbb{Z}/p\mathbb{Z}
    \to
  0
  \,.
\end{displaymath}
\end{lemma}
\begin{proof}
Consider the following [[commuting diagram]]

\begin{displaymath}
\itexarray{
     \vdots && \vdots && \vdots
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p^3\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^4 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p^2\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^3 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     \mathbb{Z}/p\mathbb{Z}
       &\overset{p\cdot (-)}{\longrightarrow}&
     \mathbb{Z}/p^2 \mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
     \\
     \downarrow
       &&
     \downarrow
       &&
     \downarrow
     \\
     0
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}
       &\longrightarrow&
     \mathbb{Z}/p\mathbb{Z}     
  }
  \,.
\end{displaymath}
Each horizontal sequence is exact. Taking the [[limit]] over the vertical sequences yields the sequence in question. Since limits commute over limits, the result follows.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{topology}{}\subsubsection*{{Topology}}\label{topology}

The ring of $p$-adic integers has the following properties:

\begin{itemize}%
\item As a [[topological space]], it is [[compact space|compact]], [[Hausdorff space|Hausdorff]], and [[totally disconnected space|totally disconnected]] (i.e., is a [[Stone space]]). Moreover, every point is an [[accumulation point]], and there is a countable basis of [[clopen set|clopen sets]] -- a Stone space with these properties must be [[homeomorphism|homeomorphic]] to [[Cantor space]].


\item As a [[topological group]] under addition, it is therefore an [[almost connected group]]. As an [[abelian group|abelian]] [[compact group]], it is [[Pontryagin duality|Pontryagin dual]] to the [[Prüfer group|Prüfer]] $p$-group as [[discrete group]].



\end{itemize}
\hypertarget{relation_to_profinite_completion_of_the_integers}{}\subsubsection*{{Relation to profinite completion of the integers}}\label{relation_to_profinite_completion_of_the_integers}

\begin{example}
\label{ProfCompletionOfIntegers}\hypertarget{ProfCompletionOfIntegers}{}
The [[profinite completion of the integers]] is

\begin{displaymath}
\widehat {\mathbb{Z}} 
    \coloneqq 
  \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z})
  \,.
\end{displaymath}
This is [[isomorphism|isomorphic]] to the [[product]] of the $p$-adic integers for all $p$

\begin{displaymath}
\widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p
  \,.
\end{displaymath}
\end{example}
(\hyperlink{Lenstra}{e.g. Lenstra, example 2.2})

\begin{defn}
\label{}\hypertarget{}{}
The ring of integral [[adeles]] $\mathbb{A}_{\mathbb{Z}}$ is the [[product]] of the profinite completion $\widehat{\mathbb{Z}}$ of the integers, example \ref{ProfCompletionOfIntegers}, with the [[real numbers]]

\begin{displaymath}
\mathbb{A}_{\mathbb{Z}}
  \coloneqq
  \mathbb{R} \times \widehat{\mathbb{Z}}
  \,.
\end{displaymath}
The [[group of units]] of the ring of adeles is called the group of [[ideles]].

\end{defn}
\hypertarget{pontrajgin_duality_to_prfer_group}{}\subsubsection*{{Pontrajgin duality to Pr\"u{}fer $p$-group}}\label{pontrajgin_duality_to_prfer_group}

Under [[Pontryagin duality]], the [[abelian group]] underlying $\mathbb{Z}_p$ maps to the [[Prüfer p-group]] $\mathbb{Z}[p^{-1}]/\mathbb{Z}$, see at \emph{[[Pontryagin duality for torsion abelian groups]]}.

\begin{displaymath}
\itexarray{
     &\mathbb{Z}[p^{-1}]/\mathbb{Z}
     &\hookrightarrow&
     \mathbb{Q}/\mathbb{Z}
     &\hookrightarrow&
     \mathbb{R}/\mathbb{Z}
     \\
     {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow
     \\
     &\mathbb{Z}_p
     &\leftarrow&
     \hat \mathbb{Z}
     &\leftarrow&
     \mathbb{Z}
  }
\end{displaymath}
\hypertarget{AsFormalNeighbourhoodOfPrime}{}\subsubsection*{{As the formal neighbourhood of a prime}}\label{AsFormalNeighbourhoodOfPrime}

The [[formal spectrum]] $Spf(\mathbb{Z}_p)$ of $\mathbb{Z}_p$ may be understood as the [[formal neighbourhood]] of the point corresponding to the [[prime]] $p$ in the [[prime spectrum]] $Spec(\mathbb{Z})$ of the [[integers]]. The inclusion

\begin{displaymath}
\{p\}
  \hookrightarrow
  Spf(\mathbb{Z}_p)
  \hookrightarrow
  Spec(\mathbb{Z})
\end{displaymath}
is the [[Isbell duality|formal dual]] of the canonical [[projection]] maps $\mathbb{Z}\to \mathbb{Z}_p\to \mathbb{Z}/(p)$.

This plays a central role for instance in the [[function field analogy]]. It is highlighted for instance in (\href{Hartl06}{Hartl 06, 1.1}, \hyperlink{Buium13}{Buium 13, section 1.1.3}). See also at \emph{[[arithmetic jet space]]} and at \emph{[[ring of Witt vectors]]}.

[[!include infinitesimal and local - table]]

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

\begin{itemize}%
\item $p$-[[p-adic number|adic number]], [[adele]].


\item [[p-adic complex number]]


\item [[Z-infinity-module]]


\item [[p-completion]]


\item [[local-global principle]]


\item [[p-adic homotopy]]


\item [[ℓ-adic cohomology]]


\item [[p-adic geometry]]


\item [[p-adic physics]]


\item [[motivic integration]]



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

Introductions and surveys include

\begin{itemize}%
\item [[Dennis Sullivan]], pp. 9 of \emph{Localization, Periodicity and Galois Symmetry} (The 1970 MIT notes) edited by [[Andrew Ranicki]], K-Monographs in Mathematics, Dordrecht: Springer (\href{http://www.maths.ed.ac.uk/~aar/surgery/gtop.pdf}{pdf})


\item Bernard Le Stum, \emph{One century of $p$-adic geometry -- From Hensel to Berkovich and beyond} talk notes, June 2012 (\href{http://www-irma.u-strasbg.fr/IMG/pdf/NotesCoursLeStum.pdf}{pdf})


\item [[Hendrik Lenstra]], \emph{Profinite groups} (\href{http://websites.math.leidenuniv.nl/algebra/Lenstra-Profinite.pdf}{pdf})



\end{itemize}
The [[synthetic differential geometry]]-aspect of the $p$-adic numbers is highlighted for instance in

\begin{itemize}%
\item [[Urs Hartl]], \emph{A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic} (\href{http://arxiv.org/abs/math/0607182}{arXiv:math/0607182})


\item [[Alexandru Buium]], \emph{Differential calculus with integers} (\href{http://arxiv.org/abs/1308.5194}{arXiv:1308.5194})



\end{itemize}
[[!redirects p-adic integer]] [[!redirects p-adic integers]]

[[!redirects adic integer]] [[!redirects adic integers]]

[[!redirects ring of p-adic integers]] [[!redirects rings of p-adic integers]]



\end{document}