proof of Ostrowski’s valuation theorem

This article proves Ostrowski’s theorem on valuations of , which states:

Theorem 1.

(Ostrowski) Over Q, every nontrivial absolute valueMathworldPlanetmathPlanetmathPlanetmath is equivalentPlanetmathPlanetmath either to ||p for some prime p, or to the usual absolute value ||.

We start with an estimation lemma:

Lemma 2.

If m,n>1 are integers and || any nontrivial absolute value on Q, then |m|max(1,|n|)logm/logn.

Proof. Write m=a0+a1n++arnr for ai,0ain-1, and with ar0. Then clearly


by the triangle inequalityMathworldMathworldPlanetmathPlanetmath; also, r<logmlogn.


|m| =|a0+a1n++arnr|(r+1)nmax(1,|n|)r

Replace m by mt for t a positive integer, and take tth roots of the resulting inequalityMathworldPlanetmath, to get


Now let t; the first two factors each approach 1, and the lemma follows.

Proof of Ostrowski’s theorem:
First assume that for every n>1 we have |n|>1. Then by the lemma, |m||n|logm/logn, so that for every m,n we have


Since this holds for every m,n>0, after reversing the roles of m,n, we see that in fact equality holds, so that for every m, |m|1/logm=c and |m|=clogm for some constant c; this absolute value is obviously equivalent to |m|=elogm.

If instead, for some n>1 we have |n|<1, then by the lemma, for every m, |m|1. Thus the absolute value is nonarchimedean. Define A={x|x|1} and let 𝔪A be the (unique) maximal ideal defined by 𝔪={x|x|<1}. Then A since |m|1 for every m, and 𝔪 is nonzero since otherwise the valuation would be trivial (we would have |m|=1 for every m). Thus 𝔪 is prime since 𝔪 is, so is equal to (p) for some rational prime p. Now, if pa for an integer a, then |a| cannot be strictly less than 1 (else it would be in (p)), so |a|=1 and aA. But given any x, we can write x=aptb with a,b prime to p, so that


so that the valuation is obviously equivalent to the p-adic valuation.

Title proof of Ostrowski’s valuation theorem
Canonical name ProofOfOstrowskisValuationTheorem
Date of creation 2013-03-22 17:58:26
Last modified on 2013-03-22 17:58:26
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Proof
Classification msc 13A18