In mathematics, Hensel's lemma, named after Kurt Hensel, is a generic name for analogues for complete commutative rings (including p-adic fields in particular) of the Newton method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively neat criteria guaranteeing a root of a polynomial. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Kurt Wilhelm Sebastian Hensel (1861–1941) was a German mathematician, a follower of Leopold Kronecker. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... The title given to this article is incorrect due to technical limitations. ... In numerical analysis, Newtons method (or the Newton-Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. ... P-adic analysis (p-adic analysis) is a branch of mathematics that deals with functions of p-adic numbers. ... Real analysis is a branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...

A version of the lemma for p-adic fields is as follows. Let f(x) be a polynomial with integer coefficients, k an integer not less than 2 and p a prime number. Suppose that r is a solution of the congruence In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... The integers are commonly denoted by the above symbol. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

If then there is a unique integer t, 0 ≤ t ≤ p-1, such that

with t defined by

If, on the other hand, and in addition, then

for all integers t.

Also, if and then has no solution for any

Generalizations

Suppose A is a commutative ring, complete with respect to an ideal , and let be a polynomial with coefficients in A. Then if a ∈ A is an "approximate root" of f in the sense that it satisfies In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

then there is an exact root b ∈ A of f "close to" a; that is,

f(b) = 0

and

Further, if f ′(a) is not a zero-divisor then b is unique.

This result has been generalized to several variables by Nicolas Bourbaki as follows: Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ...

Theorem: Let A be a commutative ring, complete with respect to an ideal m⊂ A (which is equivalent to the fact that there is an absolute value on A so that for every x in m we have |x| is strictly less than 1 and the resulting metric space is complete), and a = (a₁, …, a_n) ∈ Aⁿ an approximate solution to a system of polynomials f_i(x) ∈ A[x₁, …, x_n] in the sense that

f_i(a) ≡ 0 mod m

for 1 ≤ i ≤ n. Suppose that either det J(a) is a unit in A or that each f_i(a) ∈ (det J(a))²m, where J(a) is the Jacobian matrix of a with respect to the f_i. Then there is an exact solution b = (b₁, …, b_n) in the sense that In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

f_i(b) = 0

and furthermore this solution is "close to" a in the sense that

b_i ≡ a_i mod m

for 1 ≤ i ≤ n.

Related concepts

Completeness of a ring is not a necessary condition for the ring to have the Henselian property: Goro Azumaya in 1950 defined a commutative local ring satisfying the Henselian property for the maximal ideal m to be a Henselian ring. 1950 (MCML) was a common year starting on Sunday (link will take you to calendar). ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ...

Masayoshi Nagata proved in the 1950s that for any commutative local ring A with maximal ideal m there always exists a smallest ring A^h containing A such that A^h is Henselian with respect to mA^h. This A^h is called the Henselization of A. If A is noetherian, A^h will also be noetherian, and A^h is manifestly algebraic as it is constructed as a limit of étale neighbourhoods. This means that A^h is usually much smaller than the completion  while still retaining the Henselian property and remaining in the same category. Masayoshi Nagata is a Japanese mathematician, known for his work in the field of commutative algebra. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ...

References

• Eisenbud, David. Commutative Algebra with a View Toward Algebraic Geometry. Springer Graduate Texts in Mathematics, no. 150.
• Milne, J. S. Étale Cohomology. Princeton, 1980.