discrete valuation ring DiscreteValuationRing 2002-02-03 01:40:02 2006-01-17 21:19:45 Definition uniformizer uniformizing element order % 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{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{discrete valuation ring} $R$ is a principal ideal domain with exactly one \textbf{nonzero} maximal ideal $M$. Any generator $t$ of $M$ is called a \emph{uniformizer} or \emph{uniformizing element} of $R$; in other words, a uniformizer of $R$ is an element $t \in R$ such that $t \in M$ but $t \notin M^2$. Given a discrete valuation ring $R$ and a uniformizer $t \in R$, every element $z \in R$ can be written uniquely in the form $u \cdot t^n$ for some unit $u \in R$ and some nonnegative integer $n \in \mathbb{Z}$. The integer $n$ is called the \emph{order} of $z$, and its value is independent of the choice of uniformizing element $t \in R$.