Talk:PlanetPhysics/Gelfand Tornheim Theorem
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[edit source]%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Gelfand--Tornheim theorem %%% Primary Category Code: 02.30.-f %%% Filename: GelfandTornheimTheorem.tex %%% Version: 2 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}
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\textbf{\htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html}.}\, Any normed \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} 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 Theory of Algebraic Numbers}. \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959). \end{thebibliography}
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