Ostrowski's theorem

In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers $\mathbb {Q}$ is equivalent to either the usual real absolute value or a $p$ -adic absolute value.^[1]

^ Koblitz, Neal (1984). P-adic numbers, p-adic analysis, and zeta-functions. Graduate Texts in Mathematics (2nd ed.). New York: Springer-Verlag. p. 3. ISBN 978-0-387-96017-3. Retrieved 24 August 2012. Theorem 1 (Ostrowski). Every nontrivial norm ‖ ‖ on $\mathbb {Q}$ is equivalent to $| | p$ for some prime $p$ or for $p = \infty$ .