$p$-adic integers PAdicIntegers 2002-06-18 02:46:03 2004-07-19 15:13:01 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage[all]{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\ilim}{\,\underset{\longleftarrow}{\lim}\,} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\p}{\mathfrak{p}} \section{Basic construction} For any prime $p$, the {\em $p$--adic integers} is the ring obtained by taking the completion of the integers $\Z$ with respect to the metric induced by the norm \begin{equation}\label{valuation} |x| := \frac{1}{p^{\nu_p(x)}},\ \ x \in \Z, \end{equation} where $\nu_p(x)$ denotes the largest integer $e$ such that $p^e$ divides $x$. The induced metric $d(x,y) := |x-y|$ is called the {\em $p$--adic metric} on $\Z$. The ring of $p$--adic integers is usually denoted by $\Z_p$, and its fraction field by $\Q_p$. \section{Profinite viewpoint} The ring $\Z_p$ of $p$--adic integers can also be constructed by taking the inverse limit $$ \Z_p := \ilim \Z/p^n\Z $$ over the inverse system $\cdots \to \Z/p^2\Z \to \Z/p\Z \to 0$ consisting of the rings $\Z/p^n\Z$, for all $n \geq 0$, with the projection maps defined to be the unique maps such that the diagram $$ \xymatrix{ & \Z \ar[dl] \ar[dr]\\ \Z/p^{n+1}\Z \ar[rr] & & \Z/p^n\Z } $$ commutes. An algebraic and topological isomorphism between the two constructions is obtained by taking the coordinatewise projection map $\Z \to \ilim \Z/p^n\Z$, extended to the completion of $\Z$ under the $p$--adic metric. This alternate characterization shows that $\Z_p$ is compact, since it is a closed subspace of the space $$ \prod_{n \geq 0} \Z/p^n\Z $$ which is an infinite product of finite topological spaces and hence compact under the product topology. \section{Generalizations} If we interpret the prime $p$ as an equivalence class of valuations on $\Q$, then the field $\Q_p$ is simply the completion of the topological field $\Q$ with respect to the metric induced by any member valuation of $p$ (indeed, the valuation defined in Equation~\eqref{valuation}, extended to $\Q$, may serve as the representative). This notion easily generalizes to other fields and valuations; namely, if $K$ is any field, and $\p$ is any prime of $K$, then the $\p$--adic field $K_\p$ is defined to be the completion of $K$ with respect to any valuation in $\p$. The analogue of the $p$--adic integers in this case can be obtained by taking the subset (and subring) of $K_\p$ consisting of all elements of absolute value less than or equal to $1$, which is well defined independent of the choice of valuation representing $\p$. In the special case where $K$ is a number field, the $\p$--adic ring $K_\p$ is always a finite extension of $\Q_p$ whenever $\p$ is a finite prime, and is always equal to either $\mathbb{R}$ or $\mathbb{C}$ whenever $\p$ is an infinite prime.