Ostrowski's theorem

Last updated January 27, 2022

For a prime $p$, let $v_p(x)$ be the largest $n$ such that $p^n$ that divides $x$.

Theorem. Let $|\cdot|$ be a nontrivial absolute value on $\mathbb{Q}$. Then either

1. $|x|=s^{-v_p(x)}$ where $x\in\mathbb{R}_{>1}$ and $p$ is prime, or
2. $|x|=|x|^\alpha$, where $0<\alpha\leq 1$ and $|\cdot|$ is the standard absolute value.

The second absolute value is necessarily archimedean.