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 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

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).



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