Ostrowski's Theorem

Theorem
Every non-trivial norm on the rational numbers $\Q$ is equivalent to either:
 * the $p$-adic Norm $\norm {\, \cdot \,}_p$ for some prime $p$

or:
 * the absolute value, $\size {\, \cdot \,}$.

Proof
Let $\norm {\, \cdot \,}$ be a non-trivial norm on the rational numbers $\Q$.

Archimedean Norm Case
Let $\norm {\, \cdot \,}$ be an Archimedean norm.

Non-Archimedean Norm Case
Let $\norm {\, \cdot \,}$ be a non-Archimedean Norm.

Also see
In the same paper, published in $1918$, also proved that, up to isomorhpism, $\R$ and $\C$ are the only fields that are complete with respect to an archimedean norm.

That result is also sometimes called Ostrowski's theorem.