GelfandTornheimTheorem 2009-05-01 08:20:23 2009-05-01 08:20:23 Theorem % this is the default PlanetPhysics 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 \textbf{Theorem.}\, Any normed field is isomorphic either to the field $\mathbb{R}$ of real numbers or to the field $\mathbb{C}$ of complex numbers.\\ The {\em normed field} means here a field $K$ having a subfield $R$ isomorphic to $\mathbb{R}$ and satisfying the following: \, There is a mapping $\|\cdot\|$ from $K$ to the set of non-negative reals such that \begin{itemize} \item $\|a\| = 0$\, if and only if\, $a = 0$, \item $\|ab\| \leqq \|a\|\cdot\|b\|$, \item $\|a+b\| \leqq \|a\|+\|b\|$, \item $\|ab\| = |a|\cdot\|b\|$\, when\, $a \in R$\, and\, $b \in K$. \end{itemize} Using the Gelfand--Tornheim theorem, it can be shown that the only fields with archimedean valuation are isomorphic to subfields of $\mathbb{C}$ and that the valuation is the usual absolute value (the complex modulus) or some positive power of the absolute value. \begin{thebibliography}{8} \bibitem{artin}Emil Artin: {\em \PMlinkescapetext{Theory of Algebraic Numbers}}. \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959). \end{thebibliography}