theorem. 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
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.
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