Gelfand--Tornheim theoremGelfandTornheimTheorem2004-02-26 04:37:192008-09-06 08:38:44Theoremvaluationnormed fieldreal numberscomplex numbers% this is the default PlanetMath preamble. as your knowledge
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\newtheorem*{thmplain}{Theorem}\begin{thmplain}
\, Any normed field is isomorphic either to the field $\mathbb{R}$ of real numbers or to the field $\mathbb{C}$ of complex numbers.
\end{thmplain}
The {\em normed field} means 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 $\|a\| = 0$\, iff\, $a = 0$
\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 (modulus) or some positive power of the absolute value.
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\bibitem{artin}Emil Artin: {\em \PMlinkescapetext{Theory of Algebraic Numbers}}. \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959).
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