PlanetPhysics/Gelfand Tornheim Theorem

\htmladdnormallink{theorem {http://planetphysics.us/encyclopedia/Formula.html}.}\, 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 $\|\cdot \|$ from $K$ to the set of non-negative reals such that

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