Gelfand--Tornheim theorem

The normed field means a field $K$ having a subfield $R$ isomorphic to $\mathbb{R}$ and satisfying the following: There is a mapping $\|\cdot\|$ from $K$ to the set of non-negative reals such that

• $\|a\| = 0$ iff $a = 0$
• $\|ab\| \leqq \|a\|\cdot\|b\|$
• $\|a+b\| \leqq \|a\|+\|b\|$
• $\|ab\| = |a|\cdot\|b\|$ 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 (modulus) or some positive power of the absolute value.

Bibliography

1

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