PlanetPhysics/Gelfand Tornheim Theorem

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}.}\, Any normed field is isomorphic either to the field of real numbers or to the field of complex numbers.\\

The normed field means here a field having a subfield isomorphic to and satisfying the following: \, There is a mapping from to the set of non-negative reals such that

  • \, if and only if\, ,
  • ,
  • ,
  • \, when\, \, and\, .

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

All Sources

edit

[1]

References

edit
  1. Emil Artin: Theory of Algebraic Numbers . \,Lecture notes. \,Mathematisches Institut, G\"ottingen (1959).