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\begin{document}

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\section*{p-adic number}

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

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

[[!include higher algebra - contents]]

\hypertarget{arithmetic}{}\paragraph*{{Arithmetic}}\label{arithmetic}

[[!include arithmetic geometry - contents]]

\hypertarget{adic_numbers}{}\section*{{$p$-adic numbers}}\label{adic_numbers}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{RecollectionOfPAdicIntegers}{Recollection of the $p$-adic integers}\dotfill \pageref*{RecollectionOfPAdicIntegers} \linebreak
\noindent\hyperlink{as_an_endomorphism_ring}{As an endomorphism ring}\dotfill \pageref*{as_an_endomorphism_ring} \linebreak
\noindent\hyperlink{PAdicNumbersProper}{The $p$-adic numbers proper}\dotfill \pageref*{PAdicNumbersProper} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{basic_properties}{Basic properties}\dotfill \pageref*{basic_properties} \linebreak
\noindent\hyperlink{as_a_field_completion}{As a field completion}\dotfill \pageref*{as_a_field_completion} \linebreak
\noindent\hyperlink{Disconnectedness}{Topological disconnectedness and G-topology}\dotfill \pageref*{Disconnectedness} \linebreak
\noindent\hyperlink{pontryagin_duality}{Pontryagin duality}\dotfill \pageref*{pontryagin_duality} \linebreak
\noindent\hyperlink{function_field_analogy}{Function field analogy}\dotfill \pageref*{function_field_analogy} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

For $p$ any [[prime number]], the \emph{$p$-adic [[numbers]]} $\mathbb{Q}_p$ (or \emph{$p$-adic rational numbers}, for emphasis) are a [[field]] that [[complete field|completes]] the field of [[rational numbers]]. As such they are analogous to [[real numbers]]. A crucial difference is that the reals form an [[archimedean field]], while the $p$-adic numbers form a [[non-archimedean field]].

$p$-Adic numbers play a role in non-archimedean [[analytic geometry]] that is analogous to the role of the [[real numbers]]/[[Cartesian spaces]] in ordinary [[differential geometry]].

Moreover, as such they serve as an approximation technique in [[arithmetic geometry]] over [[prime fields]] $\mathbb{F}_p$ (see e.g. \hyperlink{Lubicz}{Lubicz}).

There have long been speculations (see the references \hyperlink{ReferencesApplications}{below}) that this must mean that $p$-adic numbers also play a central role in the description of [[physics]], see \emph{[[p-adic physics]]}.

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

We first recall the definition and construction of the [[p-adic integers]]

\begin{itemize}%
\item \hyperlink{RecollectionOfPAdicIntegers}{Recollection of the p-adic integers}

\end{itemize}
and then consider

\begin{itemize}%
\item \hyperlink{PAdicNumbersProper}{The p-adic numbers proper}

\end{itemize}
\hypertarget{RecollectionOfPAdicIntegers}{}\subsubsection*{{Recollection of the $p$-adic integers}}\label{RecollectionOfPAdicIntegers}

Let $\mathbf{Z}$ be the [[ring]] of [[integers]] and for every $q\neq 0$, $q\mathbf{Z}$ its ideal consisting of all integer multiples of $q$, and $\mathbf{Z}/q\mathbf{Z}$ the corresponding [[quotient]], the ring of residues mod $q$.

Let now $p\in \mathbf{Z}_+$ be a [[prime number]]. Then for any two positive integers $n\geq m$ there is an inclusion $p^m \mathbf{Z}\subset p^n\mathbf{Z}$ which induces the canonical homomorphism of quotients $\phi_{n,m}:\mathbf{Z}/p^n\mathbf{Z}\to \mathbf{Z}/p^m\mathbf{Z}$. These homomorphism for all pairs $n\geq m$ form a family closed under composition, and in fact a category, which is in fact a poset, and moreover a directed system of (commutative unital) rings. The \textbf{ring of [[p-adic integers]]} $\mathbf{Z}_p$ is the (inverse) [[limit]] of this directed system (inside the category of rings).

Regarding that the rings in the system are finite, it is clear that the underlying set of $\mathbf{Z}_p$ has a natural topology as a [[profinite space|profinite]] ([[Stone space|Stone]]) space and it is in particular a [[compact space|compact]] [[Hausdorff topological space|Hausdorff topological ring]]. More concretely, $\mathbf{Z}_p$ is the closed (hence compact) subspace of the cartesian product $\prod_{n} \mathbf{Z}/p^n\mathbf{Z}$ of discrete topological spaces $\mathbf{Z}/p^n\mathbf{Z}$ (which is by the [[Tihonov theorem]] compact Hausdorff) consisting of [[thread]]s, i.e. sequences of the form $x = (...,x_n,...,x_2,x_1)$ with $x_n\in p^n\mathbf{Z}$ and satisfying $\phi_{n,m}(x_n) = x_m$.

The kernel of the projection $pr_n: \mathbf{Z}_p\to\mathbf{Z}/p^n\mathbf{Z}$, $x\mapsto x_n$ to the $n$-th component (which is the corresponding projection of the limiting cone) is $p^n\mathbf{Z}_p\subset\mathbf{Z}_p$, i.e. the sequence

\begin{displaymath}
0 \longrightarrow \mathbf{Z}_p\stackrel{p^n}\longrightarrow \mathbf{Z}_p\longrightarrow \mathbf{Z}/p^n\mathbf{Z} \longrightarrow 0
\end{displaymath}
is an [[exact sequence]] of [[abelian groups]], hence also $\mathbf{Z}_p/p^n\mathbf{Z}_p\cong \mathbf{Z}/p^n\mathbf{Z}$.

An element $u$ in $\mathbf{Z}_p$ is invertible (and called a $p$-adic unit) iff $u$ is not divisible by $p$.

Let $U\subset\mathbf{Z}_p$ be the group of all invertible elements in $\mathbf{Z}_p$. Then \emph{every element $x\in \mathbf{Z}_p$ can be uniquely written as $s= u p^n$ with $n\geq 0$ and $u\in U$}. The correspondence $x\mapsto n$ defines a [[discrete valuation]] $v_p:\mathbf{Z}_p\to \mathbf{Z}\cup\{\infty\}$ called the [[p-adic valuation]] and $n$ is said to be the $p$-adic valuation of $x$. Of course, $v_p(0)=\infty$ as required by the axioms of valuation. The [[norm]] induced by the valuation is (up to equivalence) given by ${|x|}_p = p^{-v_p(x)}$, and this in turn induces a metric

\begin{displaymath}
d(x,y) = {|x-y|}_p,
\end{displaymath}
making the ring $\mathbf{Z}_p$ a [[complete metric space]] and in fact a completion of $\mathbf{Z}$, in that $d$ is a complete [[metric]], and $\mathbf{Z}$ is dense in it.

Concretely, a $p$-adic integer $x$ may be written as a base-$p$ expansion

\begin{displaymath}
x = \sum_{n \geq 0} a_n p^n
\end{displaymath}
with $a_n \in \{0, 1, \ldots, p-1\}$. Addition and multiplication are performed with [[carrying]] as in ordinary base-$p$ arithmetic, carried infinitely far to the left if $x$ is written as $\ldots a_n a_{n-1} \ldots a_1 a_0$.

\hypertarget{as_an_endomorphism_ring}{}\paragraph*{{As an endomorphism ring}}\label{as_an_endomorphism_ring}

Algebraically, the ring of $p$-adic integers is isomorphic to the endomorphism ring $\hom(\mathbf{Z}(p^\infty), \mathbf{Z}(p^\infty))$ where $\mathbf{Z}(p^\infty)$ is the [[Pruefer group|Prüfer group]] $\mathbf{Z}(p^\infty) \coloneqq \mathbb{Z}[1/p]/\mathbb{Z}$. In particular, $\mathbf{Z}(p^\infty)$ is tautologically a $\mathbf{Z}_p$-module.

Relatedly, the additive group of $p$-adic integers is [[Pontrjagin duality|Pontrjagin dual]] to $\mathbf{Z}(p^\infty)$. Observe that $\mathbf{Z}(p^\infty)$ embeds in $S^1$ as the set of all [[root of unity|roots of unity]] of order $p^n$, and that every character $\chi: \mathbf{Z}(p^\infty) \to S^1$ factors through this embedding $\mathbf{Z}(p^\infty) \hookrightarrow S^1$.

\hypertarget{PAdicNumbersProper}{}\subsubsection*{{The $p$-adic numbers proper}}\label{PAdicNumbersProper}

The \textbf{field of $p$-adic numbers} $\mathbf{Q}_p$ is the [[field of fractions]] of the [[p-adic integers]] $\mathbf{Z}_p$. The $p$-adic valuation $v_p$ extends to a discrete valuation, also denoted $v_p$ on $\mathbf{Q}_p$. Indeed, it is still true for all $x\in \mathbf{Q}_p$ that they can be uniquely written in the form $p^n u$ where $u\in U$ (the same group $U$ as before), but now one needs to allow $n\in \mathbf{Z}$. One defines the metric on $\mathbf{Q}_p$ by the same formula as for $\mathbf{Z}_p$. It appears that $\mathbf{Q}_p$ is a [[complete field]] (in particular locally compact Hausdorff) and that $\mathbf{Z}_p$ is an \emph{open} subring.

The distance $d$ satisfies the ``ultrametric'' inequality

\begin{displaymath}
d(x,z) \leq sup\{d(x,y),d(y,z)\}
\end{displaymath}
Concretely, a $p$-adic number $x$ may be written as $\sum_{n \geq k} a_n p^n$, with only finitely many negative powers of $p$ occurring. If $k \lt 0$, the expansion is conventionally displayed as

\begin{displaymath}
x = \ldots a_1 a_0.a_{-1} \ldots a_k
\end{displaymath}
with finitely many terms to the ``right'' of the ``decimal'' point. Again such expressions are added and multiplied with carrying as in ordinary arithmetic.

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

\hypertarget{basic_properties}{}\subsubsection*{{Basic properties}}\label{basic_properties}

\begin{prop}
\label{InvertiblepAdic}\hypertarget{InvertiblepAdic}{}
An element $x\in \mathbb{Z}_p$ is invertible precisely if $x_0 \neq 0$.

\end{prop}
\hypertarget{as_a_field_completion}{}\subsubsection*{{As a field completion}}\label{as_a_field_completion}

[[Ostrowski's theorem]] implies there are precisely two kinds of [[complete field|completions]] of the [[rational numbers]]: the [[real numbers]] and the $p$-adic numbers.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{([[Ostrowski's theorem]])}

Any non-trivial [[absolute value]] on the [[rational numbers]] is equivalent either to the standard real absolute value, or to the $p$-adic absolute value.

\end{theorem}
\hypertarget{Disconnectedness}{}\subsubsection*{{Topological disconnectedness and G-topology}}\label{Disconnectedness}

While the $p$-adic numbers are complete in the [[p-adic norm]], that [[topology]] is exotic: $\mathbb{Q}_p$ is a locally compact, Hausdorff, [[totally disconnected topological space]].

For that reason the naive idea of formulating [[p-adic geometry]] in analogy to [[complex analytic geometry]] as modeled on domains in $\mathbb{Q}_p^n$, regarded with their [[subspace topology]], fails (for instance there would be no [[analytic continuation]]), as also all these domains are totally disconnected.

Instead there is (\hyperlink{Tate71}{Tate 71}) a suitable [[Grothendieck topology]] on such [[affinoid domains]] -- the \emph{[[G-topology]]} -- with respect to which there is a good theory of [[non-archimedean analytic geometry]] (``[[rigid analytic geometry]]'') and hence in particular of [[p-adic geometry]]. Moreover, one may sensibly assign to an $p$-adic domain a [[topological space]] which \emph{is} well behaved (in particular locally connected and even locally contractible), this is the \emph{[[analytic spectrum]]} construction. The resulting topological space is equipped with covers by [[affinoid domain]] under the [[analytic spectrum]] are called [[Berkovich spaces]].

\hypertarget{pontryagin_duality}{}\subsubsection*{{Pontryagin duality}}\label{pontryagin_duality}

Earlier we observed that as an additive compact Hausdorff [[topological group]], the inverse limit $\mathbf{Z}_p = \lim_{\leftarrow n} \mathbb{Z}/(p^n)$ is [[Pontryagin duality|dual]] to the discrete [[Pruefer group|Prüfer group]] $\mathbf{Z}(p^\infty) \coloneqq \mathbb{Z}[1/p]/\mathbb{Z}$ that is isomorphic to a direct limit of finite cyclic groups $\lim_{\to n} \mathbb{Z}/(p^n)$. The canonical inclusion $\mathbb{Z}[1/p] \to \mathbf{Q}_p$ induces an isomorphism $\mathbf{Z}(p^\infty) \to \mathbf{Q}_p/\mathbf{Z}_p$, in fact an isomorphism of $\mathbf{Z}_p$-modules, so there is an exact sequence

\begin{displaymath}
0 \to \mathbf{Z}_p \stackrel{i}{\hookrightarrow} \mathbf{Q}_p \stackrel{q}{\to} \mathbf{Z}(p^\infty) \to 0.
\end{displaymath}
This exact sequence is Pontrjagin self-dual in the sense that the map $\mathbf{Q}_p \to \mathbf{Q}_p^\wedge$ induced from the pairing

\begin{displaymath}
\mathbf{Q}_p \times \mathbf{Q}_p \stackrel{mult}{\to} \mathbf{Q}_p \stackrel{q}{\to} \mathbb{Z}[1/p]/\mathbb{Z} \hookrightarrow \mathbb{R}/\mathbb{Z}
\end{displaymath}
fits into an isomorphism of exact sequences

\begin{displaymath}
\itexarray{
0 & \to & \mathbf{Z}_p & \stackrel{i}{\to} & \mathbf{Q}_p & \stackrel{q}{\to} & \mathbf{Z}(p^\infty) & \to & 0 \\ 
 & & \downarrow & & \downarrow & & \downarrow & & \\ 
0 & \to & \mathbf{Z}(p^\infty)^\wedge & \stackrel{q^\wedge}{\to} & \mathbf{Q}_p^\wedge & \stackrel{i^\wedge}{\to} & \mathbf{Z}_p^\wedge & \to & 0
}
\end{displaymath}
where the commutativity of the squares can be traced to the fact that $q$ is a $\mathbf{Z}_p$-module homomorphism, and where the vertical isomorphisms on left and right come from [[Pontrjagin duality]]. The middle arrow is then an isomorphism by the [[short five lemma]] for [[topological groups]], which holds by protomodularity of topological groups.

This self-duality figures in Tate's thesis; for more, see [[ring of adeles]].

\hypertarget{function_field_analogy}{}\subsubsection*{{Function field analogy}}\label{function_field_analogy}

[[!include function field analogy -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[ring of adeles]]


\item [[carrying]]


\item [[p-adic integration]]


\item [[natural number]], [[integer]], [[rational number]], [[algebraic number]], [[Gaussian number]], [[irrational number]], [[real number]]


\item [[p-adic integer]], [[p-adic complex number]]


\item [[p-adic geometry]]

\begin{itemize}%
\item [[p-adic analytic space]], [[non-commutative analytic space]]

\end{itemize}

\item [[p-adic homotopy]]


\item [[p-adic cohomology]]


\item [[p-adic physics]]

\begin{itemize}%
\item [[p-adic string theory]]

\end{itemize}


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

The $p$-adic numbers had been introduced in

\begin{itemize}%
\item [[Kurt Hensel]], \emph{\"U{}ber eine neue Begr\"u{}ndung der Theorie der algebraischen Zahlen} Jahresbericht der Deutschen Mathematiker-Vereinigung 6 (3): 83--88. (1897)

\end{itemize}
A standard reference is

\begin{itemize}%
\item [[Jean-Pierre Serre]], \emph{A course in arithmetic}, Grad. Texts in Math. \textbf{7}, Springer (1973)

\end{itemize}
Review in the context of [[p-localization|p-local]] [[homotopy theory]] is in

\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})

\end{itemize}
Review of the use of $p$-adic numbers in [[arithmetic geometry]] includes

\begin{itemize}%
\item [[David Lubicz]], \emph{An introduction to the algorithmic of $p$-adic numbers} (\href{http://www.hyperelliptic.org/tanja/conf/summerschool08/slides/p-adics.pdf}{pdf})

\end{itemize}
A formalization in [[homotopy type theory]] and there in [[Coq]] is discussed in

\begin{itemize}%
\item \'A{}lvaro Pelayo, [[Vladimir Voevodsky]], [[Michael Warren]], \emph{A preliminary univalent formalization of the p-adic numbers} (\href{http://arxiv.org/abs/1302.1207}{arXiv:1302.1207})

\end{itemize}
$p$-adic [[differential equations]] are discussed in

\begin{itemize}%
\item [[Kiran Kedlaya]], \emph{$p$-adic differential equations} (\href{http://math.ucsd.edu/~kedlaya/18.787/compiled.pdf}{pdf}, \href{http://math.ucsd.edu/~kedlaya/18.787/}{course notes})


\item [[Gilles Cristol]], \emph{Exposants $p$-adiques et solutions dans les couronnes} (\href{http://www.math.jussieu.fr/~christol/exposants.pdf}{pdf})



\end{itemize}
The development of [[rigid analytic geometry]] starts with

\begin{itemize}%
\item [[John Tate]], \emph{Rigid analytic spaces}, Invent. Math. \textbf{12}:257--289, 1971.

\end{itemize}
[[p-adic homotopy theory]] is discussed in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Rational and p-adic Homotopy Theory]]}

\end{itemize}
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