p-adic numbers
E483405
The p-adic numbers are a system of number fields that extend the rational numbers by measuring distance with respect to divisibility by a fixed prime p, playing a central role in modern number theory and arithmetic geometry.
All labels observed (4)
| Label | Occurrences |
|---|---|
| p-adic numbers canonical | 3 |
| Z_p, the ring of p-adic integers | 1 |
| p-adic Numbers, p-adic Analysis, and Zeta-Functions | 1 |
| p-adic numbers Q_p | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4962202 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: p-adic numbers Context triple: [Kurt Hensel, notableIdea, p-adic numbers]
-
A.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
B.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
C.
cyclotomic fields
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
-
D.
Kummer congruences
Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
-
E.
Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: p-adic numbers Target entity description: The p-adic numbers are a system of number fields that extend the rational numbers by measuring distance with respect to divisibility by a fixed prime p, playing a central role in modern number theory and arithmetic geometry.
-
A.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
B.
Iwasawa theory
Iwasawa theory is a branch of number theory that studies the growth of arithmetic invariants in infinite towers of number fields, particularly using p-adic methods.
-
C.
cyclotomic fields
Cyclotomic fields are number fields obtained by adjoining complex roots of unity to the rationals, playing a central role in algebraic number theory and classical geometric constructibility.
-
D.
Kummer congruences
Kummer congruences are number-theoretic relations describing how special values of Bernoulli numbers and related arithmetic functions behave modulo powers of primes, foundational in the study of p-adic L-functions and cyclotomic fields.
-
E.
Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
complete valued field
ⓘ
field ⓘ local field ⓘ non-Archimedean field ⓘ number system ⓘ topological field ⓘ |
| completionOf | rational numbers with respect to the p-adic norm ⓘ |
| constructedFrom |
p-adic absolute value
ⓘ
p-adic metric ⓘ |
| contrastsWith | real numbers ⓘ |
| definedOver | rational numbers ⓘ |
| denotedBy | Q_p ⓘ |
| distanceMeasures | divisibility by p rather than size ⓘ |
| extends | rational numbers ⓘ |
| formsPartOf | product decomposition of adeles ⓘ |
| generalizes | Hensel’s lemma applications ⓘ |
| hasAbsoluteValue | p-adic absolute value ⓘ |
| hasBasisOfNeighborhoodsOfZero | powers of p ⓘ |
| hasCharacteristic | 0 ⓘ |
| hasElementRepresentation | infinite series in powers of p with coefficients 0,…,p-1 ⓘ |
| hasPrime | p ⓘ |
| hasSubring |
Z_p
ⓘ
p-adic integers ⓘ |
| hasTopologyInducedBy | p-adic metric ⓘ |
| hasValuation | p-adic valuation ⓘ |
| introducedBy | Kurt Hensel ⓘ |
| introducedIn | 1897 ⓘ |
| isCompleteWithRespectTo | p-adic metric ⓘ |
| isLocallyCompact | true ⓘ |
| isNonArchimedean | true ⓘ |
| isTotallyDisconnected | true ⓘ |
| maximalIdealGeneratedBy | p ⓘ |
| parameterizedBy | prime number p ⓘ |
| playsCentralRoleIn | local analysis of arithmetic problems ⓘ |
| relatedTo |
Hasse principle
ⓘ
adeles ⓘ ideles ⓘ |
| residueField | finite field F_p ⓘ |
| satisfies | ultrametric inequality ⓘ |
| usedIn |
Diophantine equations
ⓘ
Galois representations ⓘ Iwasawa theory ⓘ algebraic number theory ⓘ arithmetic geometry ⓘ local class field theory ⓘ local-global principles ⓘ number theory ⓘ p-adic Hodge theory ⓘ |
| valuationRing | p-adic integers ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: p-adic numbers Description of subject: The p-adic numbers are a system of number fields that extend the rational numbers by measuring distance with respect to divisibility by a fixed prime p, playing a central role in modern number theory and arithmetic geometry.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: p-adic numbers
linked to: p-adic numbers
linked to: p-adic numbers
subject linked to:
Gertrud Hensel