Hensel’s lemma

E483408

mathematical lemma result in number theory result in p-adic analysis

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.

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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 ⓘ

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Kurt Hensel → hasEponym → Hensel’s lemma ⓘ