Hensel's lemma

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.

First form

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

:$f(r)\; equiv\; 0\; pmod\{p^\{k-1.,$

If $f\text{'}(r)\; otequiv\; 0\; pmod\{p\},$ then there is a unique integer "t", 0 ≤ "t" ≤ "p-1", such that

:$f(r\; +\; tp^\{k-1\})\; equiv\; 0\; pmod\{p^k\},$

with "t" defined by

:$tf\text{'}(r)\; equiv\; -(f(r)/p^\{k-1\})\; pmod\{p\}.,$

If, on the other hand, $f\text{'}(r)\; equiv\; 0\; pmod\{p\},$ and in addition, $f(r)\; equiv\; 0\; pmod\{p^k\},$then

:$f(r\; +\; tp^\{k-1\})\; equiv\; 0\; pmod\{p^k\},$

for all integers "t".

Also, if $f\text{'}(r)\; equiv\; 0\; pmod\{p\},$ and $f(r)\; otequiv\; 0\; pmod\{p^k\},$ then $f(x)\; equiv\; 0\; pmod\{p^k\},$ has no solution for any $x\; equiv\; r\; pmod\{p^\{k-1.,$

Example

Suppose that "p" is an odd prime number and "a" is a quadratic residue modulo "p" relative prime to "p". Then "a" has a square root in the ring of "p"-adic integers Z_p. Indeed, let "f"("x")="x"²-"a", then its derivative is 2"x", which is not zero modulo "p" for "x" not divisible by "p" (here we use that "p" is odd). By the assumption, the congruence

: $f(r)equiv\; 0\; pmod\{p\}$

has a solution "r"₁ not divisible by "p". Starting from "r"₁ and repeatedly applying Hensel's lemma, we construct a sequence of integers { "r"_i } such that

: $r\_\{i+1\}\; equiv\; r\_i\; pmod\{p^i\},\; quad\; r\_i^2\; equiv\; a\; pmod\{p^i\}.$

This sequence has a limit, a "p"-adic integer "r" such that "r"²="a". In fact, "r" is a unique square root of "a" in Z_p congruent to "r"₁ modulo "p". Conversely, if "a" is a complete square in Z_p then it is a quadratic residue mod "p". Note that the Quadratic reciprocity law allows one to easily test whether "a" is a quadratic residue mod "p", thus we get a practical way to determine which "p"-adic numbers (for "p" odd) have an integral square root, and it can be easily extended to cover the case "p"=2.

To make the example more explicit, let us consider finding the "square root of 2" (the solution to $x^2-2=0$) in the 7-adic integers. Modulo 7, we have a solution, namely 3 (we could also take 4), so $r\_1\; =\; 3$. Hensel's lemma then allows us to find $r\_2$ as follows:

:$f(r\_1)=3^2-2=7$:$f(r\_1)/p^1=7/7=1$:$f\text{'}(r\_1)=2r\_1=6$:$tf\text{'}(r)\; equiv\; -(f(r)/p^\{k-1\})\; pmod\{p\},$ that is, $tcdot\; 6\; equiv\; -1\; pmod\{7\}$:$Rightarrow\; t\; =\; 1$:$r\_2\; =\; r\_1\; +\; tp^1\; =\; 3+1\; cdot\; 7\; =\; 10\; =13\_7$

And sure enough, $10^2equiv\; 2\; pmod\{49\}$. We can then continue, and find $r\_3\; =\; 108\; =\; 213\_7$. Each time we carry out the calculation (that is, for each successive value of "k" or "i"), one more digit is added on the left to the base 7 numeral. In the 7-adic system, this sequence converges, and the limit is thus a square root of 2 in this field.

Generalizations

Suppose "A" is a commutative ring, complete with respect to an ideal $mathfrak\; m\_A$, and let $f(x)\; in\; A\; [x]$ be a polynomial with coefficients in "A". Then if "a" ∈ "A" is an "approximate root" of "f" in the sense that it satisfies

:$f(a)\; equiv\; 0\; pmod\{f\text{'}(a)^2\; ,mathfrak\; m\}$

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

:$f(b)\; =\; 0$

and

:$b\; equiv\; a\; pmod\{f\text{'}(a)\; ,mathfrak\; m\}.$

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:

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 = ("a"₁, …, "a"_"n") ∈ "A"^"n" an approximate solution to a system of polynomials "f"_"i"(x) ∈ "A" ["x"₁, …, "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 = ("b"₁, …, "b"_"n") in the sense that

:"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.

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 m"A"^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 "Â" while still retaining the Henselian property and remaining in the same category.

See also

*Hasse-Minkowski theorem
*Newton polygon

References

* | year=1995 | volume=150.
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