Talk:PlanetPhysics/Gelfand Tornheim Theorem

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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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