# Ostrowski's theorem

**Ostrowski's theorem** is a classification result in number theory stating that every nontrivial absolute value on the rational numbers \( \mathbb{Q} \) is equivalent either to the usual real absolute value \( |\cdot|_\infty \) or to a \( p \)-adic absolute value \( |\cdot|_p \) for some prime \( p \).<sup>[1](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)</sup> It was proved by [Alexander Ostrowski](https://www.edgepedia.org/alexander-ostrowski) (1893–1986), a Ukrainian-born mathematician who worked in both analysis and algebra, in a paper published in *Acta Mathematica* in 1916.<sup>[2](http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203033516611287)</sup><sup> • </sup><sup>[3](https://mathshistory.st-andrews.ac.uk//Biographies/Ostrowski/)</sup>

| Key fact | Detail |
|---|---|
| Statement | Every nontrivial absolute value on \( \mathbb{Q} \) is equivalent to \( |\cdot|_p \) for some \( p \le \infty \)<sup>[1](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)</sup> |
| Equivalence | Two absolute values are equivalent when one is a positive power of the other, equivalently when they define the same topology<sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup> |
| \( p \)-adic absolute value | \( |x|_p = p^{-v_p(x)} \), where \( v_p \) is the \( p \)-adic valuation<sup>[1](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)</sup> |
| Non-Archimedean case | \( p \)-adic absolute values satisfy the ultrametric inequality \( |x+y|_p \le \max(|x|_p, |y|_p) \)<sup>[5](https://math.stanford.edu/~ksound/Math152A10/Ostrowski.pdf)</sup> |
| Completions | The only possible completions of \( \mathbb{Q} \) are the real numbers \( \mathbb{R} \) and the \( p \)-adic numbers \( \mathbb{Q}_p \) for each prime \( p \)<sup>[6](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)</sup> |
| Original publication | Alexander Ostrowski, *Acta Mathematica* 41 (1916), pp. 271–284<sup>[2](http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203033516611287)</sup> |

## Background: absolute values on the rationals

An absolute value on \( \mathbb{Q} \) is a function \( f : \mathbb{Q} \to \mathbb{R}_{\ge 0} \) satisfying three conditions: \( f(0) = 0 \) and \( f(x) > 0 \) for \( x \ne 0 \); multiplicativity, \( f(xy) = f(x)f(y) \); and the triangle inequality, \( f(x+y) \le f(x) + f(y) \).<sup>[5](https://math.stanford.edu/~ksound/Math152A10/Ostrowski.pdf)</sup> The usual absolute value inherited from \( \mathbb{R} \), denoted \( |\cdot|_\infty \), is one example and is called archimedean.<sup>[1](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)</sup>

The \( p \)-adic examples come from unique factorization. Every nonzero rational number can be written as \( \pm \prod_q q^{e_q} \), a product over primes with uniquely determined integer exponents. The \( p \)-adic valuation \( v_p \) extracts the exponent \( e_p \), and the \( p \)-adic absolute value is defined by \( |x|_p := p^{-v_p(x)} \), with \( |0|_p = 0 \).<sup>[1](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)</sup> Under this measure, a number is small when \( p \) divides it to a high power, so \( |p|_p = 1/p \) while \( |q|_p = 1 \) for every other prime \( q \).

Equivalence is the right notion of sameness because raising an absolute value to a positive power changes its numbers but not its geometry. Two absolute values on a field are equivalent if there exists \( \alpha \in \mathbb{R}_{>0} \) with \( |x|_0 = |x|^\alpha \) for all \( x \); equivalently, they define the same metric topology, and a theorem of Ostrowski's shows that absolute values inducing the same topology are always related by such a power.<sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup>

## The theorem and its two cases

In its common form, the theorem reads: every nontrivial absolute value on \( \mathbb{Q} \) is equivalent to \( |\cdot|_p \) for some \( p \le \infty \).<sup>[1](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)</sup> Equivalently, up to equivalence the only nontrivial absolute values on \( \mathbb{Q} \) are the usual absolute value and the \( p \)-adic absolute values.<sup>[7](https://lsa.umich.edu/content/dam/math-assets/math-document/reu-documents/ugradreu/2021/D-Pentland---A-Keisling_REU21.pdf)</sup> The proof splits on the behavior of the integers.

**The archimedean case.** If \( |n| > 1 \) for some integer \( n \ge 2 \), then \( |n| > 1 \) for all \( n \ge 2 \), and a base-expansion argument shows \( |n| = n^t \) for some \( t > 0 \), so the absolute value is a power of the ordinary one.<sup>[8](https://kconrad.math.uconn.edu/blurbs/gradnumthy/ostrowskiQ.pdf)</sup> The exponent is bounded: \( |\cdot|_\infty^e \) satisfies the triangle inequality exactly when \( e \le 1 \).<sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup>

**The non-archimedean case.** If \( |n| \le 1 \) for all integers \( n \), the absolute value satisfies the stronger ultrametric inequality \( |x+y| \le \max(|x|, |y|) \).<sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup><sup> • </sup><sup>[5](https://math.stanford.edu/~ksound/Math152A10/Ostrowski.pdf)</sup> Since \( \mathbb{Q}^\times \) is generated by \( -1 \) and the primes, some prime \( p \) must have \( |p| < 1 \), and that prime is unique: if two distinct primes \( p \) and \( \ell \) both had \( |{\cdot}| < 1 \), the relation \( px + \ell y = 1 \) for suitable integers \( x, y \) would give \( 1 \le \max(|p||x|, |\ell||y|) < 1 \), a contradiction.<sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup> The absolute value is then a power of \( |\cdot|_p \), and the prime \( p \) is unique.<sup>[8](https://kconrad.math.uconn.edu/blurbs/gradnumthy/ostrowskiQ.pdf)</sup><sup> • </sup><sup>[9](https://math.iisc.ac.in/~gadgil/PfsProgs25doc/Mathlib/NumberTheory/Ostrowski.html)</sup>

The trivial absolute value, with \( f(x) = 1 \) for all \( x \ne 0 \), is excluded by the nontriviality hypothesis.<sup>[5](https://math.stanford.edu/~ksound/Math152A10/Ostrowski.pdf)</sup>

## Consequences: places and completions

An absolute value induces a metric \( d(x,y) = |x-y| \), and completing \( \mathbb{Q} \) with respect to it produces a larger field. Ostrowski's theorem therefore identifies all possible completions: the real numbers \( \mathbb{R} \) from the archimedean absolute value, and the \( p \)-adic numbers \( \mathbb{Q}_p \) from each \( p \)-adic absolute value.<sup>[6](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)</sup> In the language of algebraic number theory, the places of \( \mathbb{Q} \), defined as equivalence classes of nontrivial absolute values, are exactly the \( p \)-adic classes and the class of the usual absolute value.<sup>[10](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)</sup> The \( p \)-adic places are called finite and the archimedean place infinite.<sup>[10](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)</sup>

The classification also underlies the local-global organization of number theory: statements about integers or rationals can be tested one place at a time, over \( \mathbb{R} \) and over each \( \mathbb{Q}_p \).<sup>[6](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)</sup>

## Analogues and generalizations

For a number field \( K \) with ring of integers \( \mathcal{O}_K \), an analogous theorem holds with prime ideals in place of prime numbers: there is exactly one equivalence class of absolute values for each prime ideal \( \mathfrak{p} \subseteq \mathcal{O}_K \), given by \( |x|_\mathfrak{p} = |\mathcal{O}_K/\mathfrak{p}|^{-\mathrm{ord}_\mathfrak{p}(x)} \), plus the archimedean classes from embeddings \( K \hookrightarrow \mathbb{C} \).<sup>[6](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)</sup><sup> • </sup><sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup> For the rational function field \( \mathbb{F}_q(t) \) over a finite field, the places correspond to irreducible polynomials together with an absolute value at infinity, \( |f/g|_\infty = q^{\deg f - \deg g} \).<sup>[10](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)</sup> In the function-field case there is no canonical normalizing constant for the absolute values, unlike the \( p \)-adic case where \( |p|_p = 1/p \) is natural.<sup>[4](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)</sup>

The theorem has also been reformulated in constructive and point-free topology, where the space of absolute values is described as a space of prime ideals paired with upper reals, recovering the classical statement as a special case.<sup>[11](https://arxiv.org/abs/2308.14758)</sup> In formal mathematics, the theorem for \( \mathbb{Q} \) is proved in the Lean mathlib library as `Rat.AbsoluteValue.equiv_real_or_padic`, which states that every absolute value on \( \mathbb{Q} \) is equivalent to the standard absolute value or to a \( p \)-adic one for a unique prime \( p \).<sup>[9](https://math.iisc.ac.in/~gadgil/PfsProgs25doc/Mathlib/NumberTheory/Ostrowski.html)</sup>

## References

1. [18.785 Number Theory, Lecture #1: Absolute values on Q, MIT (2015)](https://math.mit.edu/classes/18.785/2015fa/LectureNotes1.pdf)
2. [Alexander Ostrowski, Über einige Lösungen der Funktionalgleichung ψ(x)·ψ(x)=ψ(xy), Acta Mathematica 41 (1916), 271–284](http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203033516611287)
3. [Alexander Markowich Ostrowski (1893–1986), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Ostrowski/)
4. [](https://math.stanford.edu/~conrad/676Page/handouts/ostrowski.pdf)
5. [K. Soundararajan, Ostrowski's Theorem, Math 152A, Stanford](https://math.stanford.edu/~ksound/Math152A10/Ostrowski.pdf)
6. [E. Kirsche, Local-Global Methods in Algebraic Number Theory, University of Chicago REU (2016)](https://math.uchicago.edu/~may/REU2016/REUPapers/Kirsche.pdf)
7. [D. Pentland and A. Keisling, REU paper on absolute values, University of Michigan (2021)](https://lsa.umich.edu/content/dam/math-assets/math-document/reu-documents/ugradreu/2021/D-Pentland---A-Keisling_REU21.pdf)
8. [K. Conrad, Ostrowski's Theorem for Q, University of Connecticut](https://kconrad.math.uconn.edu/blurbs/gradnumthy/ostrowskiQ.pdf)
9. [Mathlib.NumberTheory.Ostrowski documentation](https://math.iisc.ac.in/~gadgil/PfsProgs25doc/Mathlib/NumberTheory/Ostrowski.html)
10. [R. Sharifi, Algebraic Number Theory, Chapter 5: Valuations and completions, UCLA](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)
11. [M. Ng and S. Vickers, A Point-Free Look at Ostrowski's Theorem and Absolute Values, arXiv:2308.14758](https://arxiv.org/abs/2308.14758)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › p-adic numbers › p-adic valuations and absolute values*

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