Gelfand–Tornheim theorem

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 a field $K$ having a subfield $R$ isomorphic to $ℝ$ and satisfying the following:  There is a mapping $\parallel \cdot \parallel$ from $K$ to the set of non-negative reals such that

• $\parallel a\parallel =0$  iff  $a=0$

• $\parallel ab\parallel \leqq \parallel a\parallel \cdot \parallel b\parallel$

• $\parallel a+b\parallel \leqq \parallel a\parallel +\parallel b\parallel$

• $\parallel ab\parallel =|a|\cdot \parallel b\parallel$  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 $ℂ$ and that the valuation is the usual absolute value (modulus) or some positive power of the absolute value.