Complete Archimedean Valued Field is Real or Complex Numbers

Theorem
Let $\struct{k, \norm{\,\cdot\,}}$ be a complete valued field with Archimedean norm $\norm{\,\cdot\,}$.

Then either:


 * $k$ is isomorphic to the real numbers $\R$ and $\norm{\,\cdot\,}$ is equivalent to the absolute value $\size{\,\cdot\,}$ on $\R$

or


 * $k$ is isomorphic to the complex numbers $\C$ and $\norm{\,\cdot\,}$ is equivalent to the complex modulus $\size{\,\cdot\,}$ on $\C$

Also see
This result is sometimes called Ostrowski's theorem.

In the same paper, published in $1918$, also proved that every non-trivial norm on the rational numbers is equivalent to either the $p$-adic Norm for some prime $p$ or the absolute value.

It is this latter result that is more often referred to as Ostrowski's Theorem.