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. See Complete Archimedean Valued Field is Real or Complex Numbers.

That result is also sometimes called Ostrowski's theorem.