Ostrowski's theorem

Remarks

Proof idea: suppose first that $\abs{n}_* > 1$ for some positive integer $n$. Writing integers in base $b$ and letting the number of digits grow, one shows $\abs{n}_* = n^{\alpha}$ for all positive integers $n$, with a fixed $0 < \alpha \le 1$; so $\abs{\cdot}_* = \abs{\cdot}_\infty^{\alpha}$. Otherwise $\abs{n}_* \le 1$ for all integers $n$, and the integers with $\abs{n}_* < 1$ form a nonzero prime ideal $p\Z$. Then $\abs{x}_* = \abs{p}_*^{v_p(x)}$, where $v_p$ is the p-adic valuation, and this is a power of $\abs{x}_p$.

The absolute values $\abs{\cdot}_\infty$ and $\abs{\cdot}_p$ for different $p$ are pairwise inequivalent: for instance $\abs{p}_p < 1$ while $\abs{p}_q = 1$ for a prime $q \ne p$ and $\abs{p}_\infty > 1$. The hypothesis of nontriviality is needed: the trivial absolute value, equal to $1$ on all $x \ne 0$, is equivalent to none of these.

Going deeper

For a student

The theorem explains why number theory works with the real numbers and the p-adic numbers side by side: these are the completions of $\Q$ for all its places. It leads to the product formula $\abs{x}_\infty \prod_p \abs{x}_p = 1$ for $x \in \Q^\times$, and to the adele ring. For a general number field, the same method shows that every place comes from a real or complex embedding or from a nonzero prime ideal.