# P-adic valuation

The **p-adic valuation** is a function that assigns to each nonzero rational number the exponent of a fixed prime \( p \) in its factorization: for \( a = p^{m} \cdot \frac{u}{v} \) with \( u \) and \( v \) not divisible by \( p \), one sets \( \operatorname{ord}_{p}(a) = m \), and one sets \( \operatorname{ord}_{p}(0) = \infty \).<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> It measures how deeply a rational number is divisible by \( p \), and it is the algebraic core of the p-adic numbers, the completion of \( \mathbb{Q} \) in which powers of \( p \) become small.<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \operatorname{ord}_{p}(a) = m \) when \( a = p^{m} \cdot \frac{u}{v} \), with \( u, v, m \in \mathbb{Z} \) and \( u, v \not\equiv 0 \pmod p \); \( \operatorname{ord}_{p}(0) = \infty \)<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> |
| Multiplicativity | \( \operatorname{ord}_{p}(ab) = \operatorname{ord}_{p}(a) + \operatorname{ord}_{p}(b) \) for nonzero \( a, b \in \mathbb{Q} \)<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> |
| Strong triangle rule | \( \operatorname{ord}_{p}(a+b) \ge \min(\operatorname{ord}_{p}(a), \operatorname{ord}_{p}(b)) \), with equality whenever the two valuations differ<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> |
| Absolute value | \( |x|_{p} = p^{-k} \) where \( k = v_{p}(x) \), so \( |p|_{p} = p^{-1} \) and \( |0|_{p} = 0 \)<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup> |
| Completion | The completion of \( \mathbb{Q} \) with respect to the p-adic valuation is the field \( \mathbb{Q}_{p} \) of p-adic numbers<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup> |
| Classification | By Ostrowski's theorem, the places of \( \mathbb{Q} \) are exactly the p-adic absolute values for primes \( p \) together with the usual absolute value<sup>[3](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)</sup> |
| Origin | The p-adic numbers and the term "p-adic integer" are due to Kurt Hensel, who began the investigation in the last decade of the 19th century<sup>[4](https://mathshistory.st-andrews.ac.uk/Biographies//Hensel/)</sup> |

## Definition and basic properties

For a fixed prime \( p \), every nonzero rational number \( a \) can be written uniquely in the form \( a = p^{m} \cdot \frac{u}{v} \), where \( u \) and \( v \) are integers not divisible by \( p \). The exponent \( m \) is the p-adic valuation \( \operatorname{ord}_{p}(a) \); in words, it is the power of \( p \) that divides \( a \).<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> The value \( \infty \) is assigned to \( 0 \), with the convention that \( \infty \) exceeds every integer and \( \infty + a = \infty \).<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup>

The valuation satisfies three rules for nonzero \( a, b \in \mathbb{Q} \):<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup>

1. \( \operatorname{ord}_{p}(ab) = \operatorname{ord}_{p}(a) + \operatorname{ord}_{p}(b) \),
2. \( \operatorname{ord}_{p}(a+b) \ge \min(\operatorname{ord}_{p}(a), \operatorname{ord}_{p}(b)) \),
3. if \( \operatorname{ord}_{p}(a) \neq \operatorname{ord}_{p}(b) \), then \( \operatorname{ord}_{p}(a+b) = \min(\operatorname{ord}_{p}(a), \operatorname{ord}_{p}(b)) \).

The first rule says divisibility by \( p \) adds under multiplication. The second and third are the valuation form of the ultrametric inequality: a sum is at least as divisible by \( p \) as its less divisible summand, and if the summands have different valuations the sum inherits exactly the smaller one. For example, \( v_{p}(p) = 1 \) and \( v_{p}(p^{2}) = 2 \), so \( v_{p}(p + p^{2}) = 1 \).<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup> These rules also give \( v_{p}(1) = v_{p}(-1) = 0 \), hence \( v_{p}(-y) = v_{p}(y) \).<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup>

In abstract terms, a discrete rank-one valuation of a field \( K \) is a surjective map \( v : K \to \mathbb{Z} \cup \{\infty\} \) satisfying exactly these three rules, and for a prime \( p \) the map \( v_{p} : \mathbb{Q} \to \mathbb{Z} \cup \{\infty\} \) is called the p-adic valuation of \( \mathbb{Q} \).<sup>[5](https://imsc.uni-graz.at/halterko/AlgebraicNumberTh.pdf)</sup>

## The p-adic absolute value and metric

The valuation converts into a size function by exponentiating with a negative sign: \( |x|_{p} = p^{-k} \) where \( k = v_{p}(x) \), with \( |0|_{p} = 0 \).<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup> Under this size, \( |p|_{p} = p^{-1} \) while \( |1/p|_{p} = p \), so a high power of \( p \) in the numerator makes a number small rather than large.<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> The absolute value is multiplicative, \( |xy|_{p} = |x|_{p} \, |y|_{p} \), and satisfies the strong inequality \( |x + y|_{p} \le \max(|x|_{p}, |y|_{p}) \), which implies the ordinary triangle inequality as a weaker consequence.<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup>

The function \( d_{p}(a, b) = |a - b|_{p} \) is a metric on \( \mathbb{Q} \), and two rationals are close p-adically when their difference is divisible by a high power of \( p \): a sequence \( (x_{n}) \) converges to \( a \) if and only if \( \operatorname{ord}_{p}(x_{n} - a) \to \infty \).<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup> This inverts everyday intuition about size, since \( p \) lies closer to the origin than \( 1/p \) does.<sup>[1](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)</sup>

## Completion and the p-adic numbers

Completing \( \mathbb{Q} \) with respect to the p-adic valuation produces the field of p-adic numbers \( \mathbb{Q}_{p} \).<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup> Each element of \( \mathbb{Q}_{p} \) has a unique series representation

\[ x = \sum_{i=m}^{\infty} a_{i} p^{i}, \qquad m \in \mathbb{Z}, \quad 0 \le a_{i} \le p - 1, \]

and the valuation extends by \( v_{p}(x) = m \) when the first nonzero digit is \( a_{m} \); it satisfies the same three rules on all of \( \mathbb{Q}_{p} \).<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup> The elements with \( v_{p}(x) \ge 0 \) form the ring of p-adic integers \( \mathbb{Z}_{p} \), which can also be constructed as the inverse limit of the quotients \( \mathbb{Z}/p^{n}\mathbb{Z} \).<sup>[6](https://isa-afp.org/entries/Padic_Ints.html)</sup>

## Place among the absolute values of Q

[Ostrowski's theorem](https://www.edgepedia.org/ostrowskis-theorem) classifies the completions of the rationals: the places of \( \mathbb{Q} \) are exactly the equivalence classes of the p-adic absolute values, one for each prime \( p \), together with the usual absolute value.<sup>[3](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)</sup> The usual absolute value is archimedean, while every p-adic absolute value is nonarchimedean because it satisfies the max inequality rather than merely the triangle inequality.<sup>[2](https://www.tau.ac.il/~jarden/Courses/val.pdf)</sup>

## History and applications

[Kurt Hensel](https://www.edgepedia.org/kurt-hensel) (1861 to 1941) began investigating p-adic numbers in the last decade of the 19th century, motivated by analogies between number fields and function fields, where prime numbers play a role similar to linear factors \( z - c \). His invention led to the concept of a field with valuation, which influenced later mathematics broadly, and the term "p-adic integer" is his.<sup>[4](https://mathshistory.st-andrews.ac.uk/Biographies//Hensel/)</sup>

The valuation's main technical payoff is [Hensel's lemma](https://www.edgepedia.org/hensels-lemma), described by Fernando Gouvêa as the "most important algebraic property of the p-adic numbers"; it lifts a root of a polynomial modulo \( p \) to a genuine root in \( \mathbb{Z}_{p} \), provided the derivative at the approximate root is nonzero modulo \( p \).<sup>[7](https://arxiv.org/abs/1909.11342)</sup><sup> • </sup><sup>[8](https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Herwig.pdf)</sup> Because finding roots modulo \( p \) is easier than finding exact roots, the lemma supplies both existence and a numerical method.<sup>[8](https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Herwig.pdf)</sup>

The valuation also enters logic and model theory. The work of [James Ax](https://www.edgepedia.org/james-ax) and [Simon Kochen](https://www.edgepedia.org/simon-b-kochen) in the 1960s, and of Yuri Ershov, produced the first quantifier elimination results for discrete valued fields and the decidability of the field \( \mathbb{Q}_{p} \) of p-adic numbers.<sup>[9](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/266A9F68B0BBE97CC765BE30A022C442/9780521780681c9_p173-198.pdf/arithmetic-and-geometric-applications-of-quantifier-elimination-for-valued-fields.pdf)</sup>

## References

1. [Yiduan Zheng (2020). p-Adic Numbers, Qp, and Hensel's Lemma. University of Chicago REU papers.](https://math.uchicago.edu/~may/REU2020/REUPapers/Zheng%2CYiduan.pdf)
2. [Introduction to Valuation Theory, course notes, Tel Aviv University (Jarden).](https://www.tau.ac.il/~jarden/Courses/val.pdf)
3. [Romyar Sharifi. Algebraic Number Theory, Chapter 5: Valuations and completions.](https://math.ucla.edu/~sharifi/notes/algnum-ch05.html)
4. [Kurt Hensel (1861–1941), MacTutor History of Mathematics.](https://mathshistory.st-andrews.ac.uk/Biographies//Hensel/)
5. [Algebraic Number Theory, lecture notes (Halter-Koch), Section 4.1: Absolute values and valuations.](https://imsc.uni-graz.at/halterko/AlgebraicNumberTh.pdf)
6. [Hensel's Lemma for the p-adic Integers, Archive of Formal Proofs.](https://isa-afp.org/entries/Padic_Ints.html)
7. [A Formal Proof of Hensel's Lemma over the p-adic Integers, arXiv:1909.11342.](https://arxiv.org/abs/1909.11342)
8. [The p-Adic Completion of Q and Hensel's Lemma, University of Chicago REU papers (Herwig).](https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Herwig.pdf)
9. [Arithmetic and Geometric Applications of Quantifier Elimination for Valued Fields, Model Theory, Algebra, and Geometry (Cambridge University Press).](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/266A9F68B0BBE97CC765BE30A022C442/9780521780681c9_p173-198.pdf/arithmetic-and-geometric-applications-of-quantifier-elimination-for-valued-fields.pdf)

---
*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*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Oct 11, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license