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Gelfand–Tornheim theorem
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 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 nonnegative reals such that

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

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

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

$\ab\=a\cdot\b\$ 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 $\mathbb{C}$ and that the valuation is the usual absolute value (modulus) or some positive power of the absolute value.
References
 1 Emil Artin: Theory of Algebraic Numbers. Lecture notes. Mathematisches Institut, Göttingen (1959).
Defines:
normed field
Keywords:
real numbers, complex numbers
Related:
ExtensionOfKrullValuation, TopicEntryOnRealNumbers, BanachAlgebra, NormedAlgebra, ArchimedeanOrderedFieldsAreReal
Synonym:
GelfandTornheim theorem
Type of Math Object:
Theorem
Major Section:
Reference
Parent:
Groups audience:
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