RayClassGroup 2002-07-11 11:32:52 2002-07-11 11:32:52 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{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\p}{{\mathfrak{p}}} \newcommand{\m}{{\mathfrak{m}}} \renewcommand{\P}{{\mathfrak{P}}} \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\N}{\mathbb{N}} \renewcommand{\H}{\mathcal{H}} \newcommand{\A}{\mathbb{A}} \renewcommand{\c}{\mathcal{C}} \renewcommand{\O}{\mathcal{O}} \newcommand{\D}{\mathcal{D}} \newcommand{\lra}{\longrightarrow} \renewcommand{\div}{\mid} \newcommand{\res}{\operatorname{res}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\id}{\operatorname{id}} \newcommand{\diff}{\operatorname{diff}} \newcommand{\incl}{\operatorname{incl}} \newcommand{\Hom}{\operatorname{Hom}} \renewcommand{\Re}{\operatorname{Re}} \newcommand{\intersect}{\cap} \newcommand{\union}{\cup} \newcommand{\bigintersect}{\bigcap} \newcommand{\bigunion}{\bigcup} \newcommand{\ilim}{\,\underset{\longleftarrow}{\lim}\,} Let $\m$ be a modulus for a number field $K$. The {\em ray class group} of $K$ mod $\m$ is the group $\mathbb{I}^\m/K_{\m,1}$, where \begin{itemize} \item $\mathbb{I}^\m$ is the subgroup of the ideal group of $K$ generated by all prime ideals which do not occur in the factorization of $\m$. \item $K_{\m,1}$ is the subgroup of $\mathbb{I}^\m$ consisting of all principal ideals in the ring of integers of $K$ having the form $(\alpha)$ where $\alpha$ is multiplicatively congruent to $1 \bmod \m$. \end{itemize}