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 (1)
| Label | Occurrences |
|---|---|
| Hensel’s lemma canonical | 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.
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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 NERFINISHED ⓘ strong form of Hensel’s lemma ⓘ weak form of Hensel’s lemma NERFINISHED ⓘ |
| historicalPeriod | late 19th century ⓘ |
| introducedConcept | systematic use of p-adic methods in number theory ⓘ |
| namedAfter | Kurt Hensel NERFINISHED ⓘ |
| relatedTo |
Newton’s method
NERFINISHED
ⓘ
Q_p, the field of p-adic numbers ⓘ Z_p, the ring of p-adic integers NERFINISHED ⓘ 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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.