Ostrowski's theorem (1916)

Sketch. Let $|\cdot|$ be a nontrivial absolute value on $\mathbb{Q}$. Case 1 (non-Archimedean): if $|n| \le 1$ for every integer $n$, the set $\mathfrak{p} = \{n \in \mathbb{Z} : |n| < 1\}$ is a prime ideal of $\mathbb{Z}$ (it is closed under addition by the ultrametric inequality, which any absolute value with $|n|\le1$ on $\mathbb{Z}$ automatically satisfies, and under multiplication by primality), hence $\mathfrak{p} = (p)$ for a unique prime $p$; comparing $|p|$ to $p^{-1}$ and using multiplicativity shows $|\cdot|$ is equivalent to $|\cdot|_p$. Case 2 (Archimedean): if $|n_0| > 1$ for some integer $n_0$, write any integer $n > 1$ in base $n_0$ and use the triangle inequality together with $|n_0^k| = |n_0|^k \to \infty$ to bound $|n|$ above and below by powers of $n$ itself, forcing $|n| = n^c$ for a constant $c \in (0,1]$ independent of $n$; this makes $|\cdot|$ equivalent to $|\cdot|_\infty$.