Gelfand–Tornheim theorem


Any normed field is isomorphic either to the field of real numbers or to the field of complex numbersMathworldPlanetmathPlanetmath.

The normed field means a field K having a subfieldMathworldPlanetmath R isomorphic to and satisfying the following:   There is a mapping from K to the set of non-negative reals such that

  • a=0  iff  a=0

  • abab

  • a+ba+b

  • ab=|a|b  when  aR  and  bK

Using the Gelfand–Tornheim theorem, it can be shown that the only fields with archimedean valuation are isomorphic to subfields of and that the valuationMathworldPlanetmath is the usual absolute valueMathworldPlanetmathPlanetmathPlanetmath (modulus) or some positive power of the absolute value.


  • 1 Emil Artin: .  Lecture notes.  Mathematisches Institut, Göttingen (1959).
Title Gelfand–Tornheim theorem
Canonical name GelfandTornheimTheorem
Date of creation 2013-03-22 14:11:49
Last modified on 2013-03-22 14:11:49
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 40
Author pahio (2872)
Entry type Theorem
Classification msc 12J05
Synonym Gelfand-Tornheim theorem
Related topic ExtensionOfKrullValuation
Related topic TopicEntryOnRealNumbers
Related topic BanachAlgebra
Related topic NormedAlgebra
Related topic ArchimedeanOrderedFieldsAreReal
Defines normed field