HermitesTheorem 2005-02-24 08:35:07 2006-09-26 14:16:28 Corollary Minkowski's constant unramified outside a set of primes % 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{amsthm} \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 \newtheorem{thm}{Theorem} \newtheorem*{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem*{cor}{Corollary} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Rats}{\mathbb{Q}} The following is a corollary of Minkowski's theorem on ideal classes, which is a corollary of Minkowski's theorem on lattices. \begin{defn} Let $S=\{p_1,\ldots,p_r\}$ be a set of rational primes $p_i \in \Ints$. We say that a number field $K$ is {\bf unramified outside $S$} if any prime not in $S$ is unramified in $K$. In other words, if $p$ is ramified in $K$, then $p\in S$. In other words, the only primes that divide the discriminant of $K$ are elements of $S$. \end{defn} \begin{cor}[Hermite's Theorem] Let $S=\{p_1,\ldots,p_r\}$ be a set of rational primes $p_i \in \Ints$ and let $N\in \Nats$ be arbitrary. There is only a finite number of fields $K$ which are unramified outside $S$ and bounded degree $[K:\Rats]\leq N$. \end{cor}