Gelfand–Tornheim theorem
Theorem.
Any normed field is isomorphic either to the field $\mathbb{R}$ of real numbers or to the field $\u2102$ of complex numbers^{}.
The normed field means a field $K$ having a subfield^{} $R$ isomorphic to $\mathbb{R}$ and satisfying the following: There is a mapping $\parallel \cdot \parallel $ from $K$ to the set of nonnegative reals such that

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$\parallel a\parallel =0$ iff $a=0$

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$\parallel ab\parallel \leqq \parallel a\parallel \cdot \parallel b\parallel $

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$\parallel a+b\parallel \leqq \parallel a\parallel +\parallel b\parallel $

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$\parallel ab\parallel =a\cdot \parallel b\parallel $ when $a\in R$ and $b\in K$
Using the Gelfand–Tornheim theorem, it can be shown that the only fields with archimedean valuation are isomorphic to subfields of $\u2102$ and that the valuation^{} is the usual absolute value^{} (modulus) or some positive power of the absolute value.
References
 1 Emil Artin: . Lecture notes. Mathematisches Institut, Göttingen (1959).
Title  Gelfand–Tornheim theorem 
Canonical name  GelfandTornheimTheorem 
Date of creation  20130322 14:11:49 
Last modified on  20130322 14:11:49 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  40 
Author  pahio (2872) 
Entry type  Theorem 
Classification  msc 12J05 
Synonym  GelfandTornheim theorem 
Related topic  ExtensionOfKrullValuation 
Related topic  TopicEntryOnRealNumbers 
Related topic  BanachAlgebra 
Related topic  NormedAlgebra 
Related topic  ArchimedeanOrderedFieldsAreReal 
Defines  normed field 