# P-adic analysis
P-adic analysis is the branch of analysis carried out over the fields of p-adic numbers \( \mathbb{Q}_p \), obtained by completing the rational numbers \( \mathbb{Q} \) with respect to the p-adic absolute value instead of the usual real absolute value. It supplies a calculus and function theory adapted to questions of divisibility by a fixed prime \( p \), and it underlies much of modern number theory, from local class field theory to p-adic [Hodge theory](https://www.edgepedia.org/hodge-theory).
| Key fact | Detail |
|---|---|
| Absolute value | For a prime \( p \), \( \|a\|_p = p^{-v_p(a)} \) for \( a \neq 0 \), where \( v_p(a) \) is the exponent of \( p \) in \( a \); thus numbers are close when their difference is divisible by a high power of \( p \)[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch05.html) |
| Classification | Up to equivalence, and except for the trivial absolute value, the absolute values on \( \mathbb{Q} \) are exactly the usual one and one \( \|\cdot\|_p \) for each prime \( p \) (Ostrowski's theorem)[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf) • [1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch05.html) |
| The fields | \( \mathbb{Q}_p \) is the completion of \( \mathbb{Q} \) for \( \|\cdot\|_p \); its valuation ring \( \mathbb{Z}_p = \{a \in \mathbb{Q}_p : \|a\|_p \le 1\} \) is a discrete valuation ring with maximal ideal \( p\mathbb{Z}_p \)[3](https://homepage.univie.ac.at/dietrich.burde/papers/burde_71_ant.pdf) |
| Topology | \( \|\cdot\|_p \) is non-archimedean (the integers form a bounded set), and the induced topology is totally disconnected[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf) |
| Local fields | A p-adic local field is a finite extension \( K \) of \( \mathbb{Q}_p \); the absolute value extends uniquely by \( \|x\| = \|N_{K/\mathbb{Q}_p}(x)\|_p^{1/[K:\mathbb{Q}_p]} \), making \( K \) complete and locally compact[4](https://www.math.uni-bonn.de/people/huybrech/Magni.pdf) |
| Complex counterpart | The algebraic closure \( \overline{\mathbb{Q}}_p \) is not complete; its completion \( \mathbb{C}_p \) is the smallest extension of \( \mathbb{Q} \) that is both algebraically closed and complete for the p-adic absolute value[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf) |
| Origin | Kurt Hensel created the p-adic numbers around 1899, motivated by Weierstrass's power-series calculus; Helmut Hasse proved the local-global principle for quadratic forms in his 1921 dissertation[5](https://link.springer.com/article/10.1365/s13291-022-00262-8) |
## The p-adic absolute value and completion
An absolute value on a field \( K \) is a map \( \|\cdot\| : K \to \mathbb{R}_{\ge 0} \) that is zero only at zero, multiplicative, and satisfies a triangle inequality. For each prime \( p \), the p-adic valuation \( v_p \) counts the power of \( p \) dividing a rational number, and the p-adic absolute value is defined by \( \|0\|_p = 0 \) and \( \|a\|_p = p^{-v_p(a)} \)[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch05.html). Under this measure, \( \|p\|_p = 1/p \), so a number is small precisely when it is highly divisible by \( p \). [Ostrowski's theorem](https://www.edgepedia.org/ostrowskis-theorem) shows that these absolute values, together with the usual real one, exhaust the nontrivial absolute values on \( \mathbb{Q} \) up to equivalence[1](https://www.math.ucla.edu/~sharifi/notes/algnum-ch05.html) • [2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf).
Completing \( \mathbb{Q} \) with respect to \( \|\cdot\|_p \), in exact analogy with the construction of \( \mathbb{R} \) from Cauchy sequences, produces the field \( \mathbb{Q}_p \) of p-adic numbers[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf) • [6](https://androma.org/page/Cambridge%20III%20Local%20Fields). Its elements can be written as expansions \( a_{-n}p^{-n} + \cdots + a_{-1}p^{-1} + a_0 + a_1 p + a_2 p^2 + \cdots \) with digits \( 0 \le a_i < p \); those with no negative-index terms form the ring \( \mathbb{Z}_p \) of p-adic integers[3](https://homepage.univie.ac.at/dietrich.burde/papers/burde_71_ant.pdf). Equivalently, \( \mathbb{Z}_p \) arises as the projective limit of the residue class rings \( \mathbb{Z}/p^n\mathbb{Z} \)[5](https://link.springer.com/article/10.1365/s13291-022-00262-8).
The completion is genuinely needed: \( \mathbb{Q} \) is not complete under any \( \|\cdot\|_p \). For \( p = 5 \), one can construct a sequence of integers \( a_n \) with \( a_n^2 \equiv 6 \pmod{5^{n+1}} \) and \( a_{n+1} \equiv a_n \pmod{5^n} \); this is Cauchy for \( \|\cdot\|_5 \), and a limit \( d \) in \( \mathbb{Q} \) would satisfy \( d^2 = 6 \), which is impossible[7](https://www.mathstat.dal.ca/~yanghs/course-notes/p-adic%20analysis.pdf).
## Non-archimedean features
The p-adic absolute value is non-archimedean: it satisfies the strong triangle inequality \( \|x + y\|_p \le \max(\|x\|_p, \|y\|_p) \). A visible consequence is that the integers, unbounded in the usual metric, form a bounded subset of \( \mathbb{Q} \) under \( \|\cdot\|_p \)[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf).
The p-adic topology is also totally disconnected, which blocks a direct transfer of parts of classical analysis such as line integrals. One remedy, due to Vladimir Berkovich, embeds \( \mathbb{C}_p \) into a larger path-connected analytic space, now called a [Berkovich space](https://www.edgepedia.org/berkovich-space)[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf).
## Local fields and the p-adic complex numbers
A p-adic local field is a finite extension \( K \) of \( \mathbb{Q}_p \). The absolute value \( \|\cdot\|_p \) extends to \( K \) uniquely, explicitly by \( \|x\| = \|N_{K/\mathbb{Q}_p}(x)\|_p^{1/[K:\mathbb{Q}_p]} \) using the field norm, and this extension makes \( K \) a complete, locally compact topological field[4](https://www.math.uni-bonn.de/people/huybrech/Magni.pdf). Local compactness is what allows measure theory and harmonic analysis on \( \mathbb{Q}_p \), paralleling the role of \( \mathbb{R} \) and \( \mathbb{C \) in classical analysis.
Unlike \( \mathbb{R} \), whose algebraic closure \( \overline{\mathbb{R}} = \mathbb{C} \) is already complete, the algebraic closure \( \overline{\mathbb{Q}}_p \) is not complete. Completing it yields \( \mathbb{C}_p \), the smallest extension of \( \mathbb{Q} \) that is algebraically closed and complete for the p-adic absolute value[2](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf).
## History
[Kurt Hensel](https://www.edgepedia.org/kurt-hensel), a student of Leopold Kronecker in Berlin and familiar with Karl Weierstrass's power-series method, conceived the p-adic numbers around 1899 as an analog of power-series expansion for algebraic numbers[5](https://link.springer.com/article/10.1365/s13291-022-00262-8) • [8](https://www.math.berlin/mathematiker/kurt-hensel.html). The construction initially met with little acceptance; [Helmut Hasse](https://www.edgepedia.org/helmut-hasse), Hensel's doctoral student, gave it decisive weight by proving in his 1921 dissertation the local-global principle Hensel had anticipated, that global statements about quadratic forms over \( \mathbb{Q} \) can be derived from the local p-adic and real completions[5](https://link.springer.com/article/10.1365/s13291-022-00262-8) • [8](https://www.math.berlin/mathematiker/kurt-hensel.html).
## Modern developments
P-adic analysis now supports several large theories. Tate's 1971 work on rigid analytic spaces provided a framework for p-adic analytic varieties, and in his 1967 paper on p-divisible groups Tate initiated the p-adic Hodge theory of such varieties, conjecturing a Hodge-like decomposition for p-adic étale cohomology; Fontaine refined the conjecture, and proofs accumulated between 1985 and 2011. In 2011 [Peter Scholze](https://www.edgepedia.org/peter-scholze) proved Tate's conjecture using perfectoid methods, opening a broad subsequent development[9](https://arxiv.org/abs/2601.16557). On the representation-theoretic side, p-adic analogs of the differential-equation approach to [Lie group](https://www.edgepedia.org/lie-group) representations have been developed through Berthelot's theory of arithmetic D-modules and the Schneider–Stuhler theory of sheaves on buildings[10](https://journal.hep.com.cn/fmc/EN/10.1007/s11464-022-1008-z).
## References
1. [Romyar Sharifi, Algebraic Number Theory, Chapter 5: Valuations and completions, UCLA](https://www.math.ucla.edu/~sharifi/notes/algnum-ch05.html)
2. [Peter Stoll, p-adic Analysis in Arithmetic Geometry, lecture notes, University of Bayreuth](https://mathe2.uni-bayreuth.de/stoll/teaching/pAdicAnalysis-WS2015/Skript-pAdicAnalysis-pub-print.pdf)
3. [Dietrich Burde, Algebraic Number Theory, University of Vienna](https://homepage.univie.ac.at/dietrich.burde/papers/burde_71_ant.pdf)
4. [Fundamental p-adic analysis, lecture notes, University of Bonn](https://www.math.uni-bonn.de/people/huybrech/Magni.pdf)
5. [Zum Einhundertsten Geburtstag des Lokal-Global-Prinzips in der Zahlentheorie, Jahresbericht der DMV](https://link.springer.com/article/10.1365/s13291-022-00262-8)
6. [Cambridge III Local Fields notes (annotated)](https://androma.org/page/Cambridge%20III%20Local%20Fields)
7. [PMATH 940: p-adic Analysis Notes, Dalhousie University](https://www.mathstat.dal.ca/~yanghs/course-notes/p-adic%20analysis.pdf)
8. [Mathematiker des Monats Juni 2017: Kurt Hensel, Berliner Mathematische Gesellschaft](https://www.math.berlin/mathematiker/kurt-hensel.html)
9. [Pierre Colmez and Wiesława Nizioł, Hodge Theory of p-adic analytic varieties: a survey, arXiv](https://arxiv.org/abs/2601.16557)
10. [Differential equations and Lie group representations, Frontiers of Mathematics in China](https://journal.hep.com.cn/fmc/EN/10.1007/s11464-022-1008-z)
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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 analysis*
*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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