Hensel’s lemma
E483408
Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Hensel lifting | 3 |
| Hensel’s lemma canonical | 2 |
| Hensel lemma | 1 |
| multivariate Hensel’s lemma | 1 |
| weak form of Hensel’s lemma | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4962225 — 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: Hensel’s lemma Context triple: [Kurt Hensel, hasEponym, Hensel’s lemma]
-
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.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
C.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
D.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
E.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
- 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: Hensel’s lemma Target entity description: Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
-
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.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
C.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
D.
Bateman–Horn conjecture
The Bateman–Horn conjecture is a far-reaching unproven statement in number theory that predicts how often sets of polynomial expressions simultaneously take prime values, generalizing several earlier conjectures about the distribution of prime numbers.
-
E.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical lemma
ⓘ
result in number theory ⓘ result in p-adic analysis ⓘ |
| appliesTo |
polynomial congruences modulo powers of a prime
ⓘ
polynomials over p-adic integers ⓘ |
| assumption |
existence of a solution modulo p^k for some k
ⓘ
polynomial with coefficients in Z_p or Z ⓘ prime number p ⓘ |
| characterizes | Henselian local rings ⓘ |
| conclusion |
existence of a root in Z_p corresponding to a compatible system of roots modulo p^n
ⓘ
existence of a unique lift of a root modulo higher powers of p ⓘ |
| condition |
f(a) ≡ 0 (mod p) and f′(a) not ≡ 0 (mod p)
ⓘ
non-vanishing derivative modulo p ⓘ or stronger divisibility conditions on f(a) and f′(a) ⓘ |
| coreConcept |
lifting solutions modulo p^n to solutions modulo p^{n+1}
ⓘ
p-adic analogue of Newton’s method ⓘ |
| field |
number theory
ⓘ
p-adic analysis ⓘ |
| generalizationOf | root lifting from modulo p to modulo p^n ⓘ |
| hasVariant |
Henselian ring criterion
ⓘ
multivariate Hensel’s lemma ⓘ
linked to:
Hensel’s lemma
strong form of Hensel’s lemma ⓘ weak form of Hensel’s lemma ⓘ
linked to:
Hensel’s lemma
|
| historicalPeriod | late 19th century ⓘ |
| introducedConcept | systematic use of p-adic methods in number theory ⓘ |
| namedAfter | Kurt Hensel ⓘ |
| relatedTo |
Newton’s method
ⓘ
Q_p, the field of p-adic numbers ⓘ Z_p, the ring of p-adic integers ⓘ
linked to:
p-adic numbers
implicit function theorem over non-Archimedean fields ⓘ lifting idempotents in complete local rings ⓘ local fields ⓘ p-adic numbers ⓘ p-adic valuation ⓘ |
| usedFor |
computing p-adic approximations of roots
ⓘ
constructing p-adic integers as limits of solutions modulo p^n ⓘ lifting factorizations modulo p to factorizations over Z_p ⓘ lifting roots of polynomials modulo a prime to p-adic roots ⓘ local-global principles in number theory ⓘ proving existence of roots in Q_p ⓘ solving Diophantine equations locally ⓘ studying factorization of polynomials over p-adic fields ⓘ |
| usedIn |
algebraic number theory
ⓘ
algorithmic number theory ⓘ computational algebra systems ⓘ local class field theory ⓘ |
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: Hensel’s lemma Description of subject: Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hensel’s lemma
linked to: Hensel’s lemma
linked to: Hensel’s lemma
linked to: Hensel’s lemma