Modulus 2002-04-16 22:14:44 2002-04-16 22:14:44 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}} A {\em modulus} for a number field $K$ is a formal product $$ \prod_{\p} \p^{n_\p} $$ where \begin{itemize} \item The product is taken over all finite primes and infinite primes of $K$ \item The exponents $n_\p$ are nonnegative integers \item All but finitely many of the $n_\p$ are zero \item For every real prime $\p$, the exponent $n_\p$ is either 0 or 1 \item For every complex prime $\p$, the exponent $n_\p$ is 0 \end{itemize} A modulus can be written as a product of its finite part $$ \prod_{\p \text{ finite}} \p^{n_\p} $$ and its infinite part $$ \prod_{\p \text{ real}} \p^{n_\p}, $$ with the finite part equal to some ideal in the ring of integers $\mathcal{O}_K$ of $K$, and the infinite part equal to the product of some subcollection of the real primes of $K$.