theorem. Any normed field is isomorphic either to the field ℝ of real numbers or to the field ℂ of complex numbers.

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

• ∥a∥ = 0 if and only if a = 0,
• ∥ab∥≦∥a∥⋅∥b∥,
• ∥a + b∥≦∥a∥ + ∥b∥,
• ∥ab∥ = |a|⋅∥b∥ when a ∈ R and b ∈ K.

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

References