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Gelfand--Tornheim theorem (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 here a field $K$ having a subfield $R$ isomorphic to $\mathbb{R}$ and satisfying the following:   There is a mapping $\Vert\cdot\Vert$ from $K$ to the set of non-negative reals such that

  • $\Vert a\Vert = 0$  if and only if  $a = 0$,
  • $\Vert ab\Vert \leqq \Vert a\Vert\cdot\Vert b\Vert$,
  • $\Vert a+b\Vert \leqq \Vert a\Vert+\Vert b\Vert$,
  • $\Vert ab\Vert = \vert a\vert\cdot\Vert b\Vert$  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 (the complex modulus) or some positive power of the absolute value.

Bibliography

1
Emil Artin: Theory of Algebraic Numbers.  Lecture notes.  Mathematisches Institut, Göttingen (1959).



"Gelfand--Tornheim theorem" is owned by pahio.

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Other names:  Gelfand-Tornheim theorem

Cross-references: field, theorem

This is version 2 of Gelfand--Tornheim theorem, born on 2009-05-01, modified 2009-05-02.
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Physics Classification: 02.30.-f (Function theory, analysis)
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