SiegelKlingenTheorem 2006-06-20 12:32:15 2006-06-20 12:32:15 Theorem Dedekind zeta function % 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} \theoremstyle{definition} \newtheorem{exa}{Example} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Rats}{\mathbb{Q}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\Cl}{\operatorname{Cl}} \begin{thm}[Siegel-Klingen Theorem, \cite{klingen},\cite{siegel}] Let $K$ be a totally real number field and let $\zeta(s,K)$ be the Dedekind zeta function of $K$. If $n\geq 1$ is an integer then $\zeta(-n,K)$ is a rational number (i.e. $\zeta(-n,K)\in \Rats$). \end{thm} \begin{thebibliography}{99} \bibitem{klingen} Klingen, Helmut, {\em \"Uber die Werte der Dedekindschen Zetafunktion}. (German) Math. Ann. 145 1961/1962 265--272. \bibitem{siegel} Siegel, Carl Ludwig, {\em \"Uber die analytische Theorie der quadratischen Formen. III}. (German) Ann. of Math. (2) 38 (1937), no. 1, 212--291. \end{thebibliography}