Hensel's lemma

The henselsche Lemma (after Kurt Hensel ) is a statement from the mathematical branch of algebra .

It was proven by Theodor Schönemann as early as 1846 before Hensel .

formulation

Let it be a complete, non-Archimedean evaluated body with evaluation ring and residual class field . If there is a polynomial whose reduction is the product of two prime polynomials , there are polynomials such that and or is the reduction of or is. ${\displaystyle K}$ ${\displaystyle A}$ ${\displaystyle k}$ ${\displaystyle f\in A[X]}$ ${\displaystyle {\bar {f}}\in k[X]}$ ${\displaystyle {\bar {g}},{\bar {h}}\in k[X]}$ ${\displaystyle g,h\in A[X]}$ ${\displaystyle f=gh}$ ${\displaystyle {\bar {g}}}$ ${\displaystyle {\bar {h}}}$ ${\displaystyle g}$ ${\displaystyle h}$

Examples

• With Hensel's lemma one can show that the field of the p -adic numbers contains the -th roots of unity : ${\displaystyle (p-1)}$

For a prime number let the field of -adic numbers, and . The polynomial is broken down into linear factors ${\displaystyle p}$ ${\displaystyle K=\mathbb {Q} _{p}}$ ${\displaystyle p}$ ${\displaystyle A=\mathbb {Z} _{p}}$ ${\displaystyle k=\mathbb {F} _{p}}$ ${\displaystyle f(X):=X^{p-1}-1}$ ${\displaystyle k}$

${\displaystyle {\bar {f}}(X)=X^{p-1}-{\bar {1}}=(X-{\bar {1}})(X-{\bar {2}})\cdots (X-{\overline {p-1}})}$.

So there are polynomials such that ${\displaystyle g_{1},\ldots ,g_{p-1}\in \mathbb {Z} _{p}[X]}$

${\displaystyle f=g_{1}\cdots g_{p-1},\qquad g_{i}(X)\equiv X-i\mod p}$

applies. The polynomials necessarily have the form with , so one can assume that H. there are so that ${\displaystyle g_{i}}$ ${\displaystyle g(X)=aX+b}$ ${\displaystyle a\in 1+p\mathbb {Z} _{p}}$ ${\displaystyle a=1}$ ${\displaystyle \zeta _{1},\ldots ,\zeta _{p-1}\in \mathbb {Z} _{p}}$

${\displaystyle X^{p-1}-1=(X-\zeta _{1})\cdots (X-\zeta _{p-1})}$

applies. The are the st roots of unity, and they can always be arranged so that . ${\displaystyle \zeta _{1},\ldots ,\zeta _{p-1}}$ ${\displaystyle (p-1)}$ ${\displaystyle \zeta _{i}\equiv i\mod p}$

• If the number is prime , then there is a with after the above . ${\displaystyle p\equiv 1\mod 4}$ ${\displaystyle \eta \in \mathbb {Q} _{p}}$ ${\displaystyle \eta ^{2}=-1}$

Because among the -th roots of unity there is one, denoted by, that creates the cyclic group of -th roots of unity . With results . ${\displaystyle (p-1)}$ ${\displaystyle \zeta }$ ${\displaystyle (p-1)}$ ${\displaystyle \eta :=\zeta ^{\tfrac {p-1}{4}}}$ ${\displaystyle \eta ^{2}=-1}$

• In the field of p -adic numbers, 0 can be represented by a non-trivial sum of squares. This means that -1 can be represented by a sum of squares and cannot be arranged . ${\displaystyle \mathbb {Q} _{p}}$ ${\displaystyle \mathbb {Q} _{p}}$

A distinction must be made between two cases:

1. ${\displaystyle p>2}$: According to the four-squares theorem there are 4 summands with . Now is a square remainder . So there is a with or and coprime . According to Hensel's lemma there is a with such that the sum of 4 non-vanishing squares is. ${\displaystyle n_{1},n_{2},n_{3},n_{4}\in \mathbb {N} }$ ${\displaystyle n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+n_{4}^{2}=p}$ ${\displaystyle n:=n_{4}^{2}-p}$ ${\displaystyle n_{4}^{2}\mod p}$ ${\displaystyle \mu \in \mathbb {F} _{p}}$ ${\displaystyle \mu ^{2}={\bar {n}}}$ ${\displaystyle (X-\mu )(X+\mu )=X^{2}-{\bar {n}}}$ ${\displaystyle (X-\mu ),(X+\mu )}$ ${\displaystyle m\in \mathbb {Z} _{p}}$ ${\displaystyle m^{2}=n=n_{4}^{2}-p}$ ${\displaystyle n_{1}^{2}+n_{2}^{2}+n_{3}^{2}+m^{2}=0}$
2. ${\displaystyle p=2}$: In the case of square roots, the Hensel's lemma cannot be used directly because of the lack of coprime numbers of the polynomials . However, with the procedure in the proof of the same it can be shown that . Be with me . For now be such that . Since is divisible by 2, we can form. Then is . Thus there is an in convergent sequence with . The sum of 5 squares disappears. ${\displaystyle \mathbb {F} _{2}}$ ${\displaystyle {\sqrt {-7}}\in \mathbb {Q} _{2}}$ ${\displaystyle m_{3}:=1\in \mathbb {Z} }$ ${\displaystyle m_{3}^{2}\equiv -7\;\mod 2^{3}}$ ${\displaystyle i=3,4,...}$ ${\displaystyle m_{i}\in \mathbb {Z} }$ ${\displaystyle m_{i}^{2}\equiv -7\mod 2^{i}}$ ${\displaystyle m_{i}^{2}+7}$
       ${\displaystyle m_{i+1}:\equiv m_{i}-{\tfrac {m_{i}^{2}+7}{2m_{i}}}\mod 2^{i+1}}$
   
       ${\displaystyle m_{i+1}^{2}\equiv m_{i}^{2}-(m_{i}^{2}+7)\equiv -7\mod 2^{i+1}}$
   ${\displaystyle \mathbb {Q} _{2}}$ ${\displaystyle m:=\lim _{i\to \infty }m_{i}}$ ${\displaystyle m^{2}=-7}$ ${\displaystyle 1^{2}+1^{2}+1^{2}+2^{2}+m^{2}=7+m^{2}}$

• Let it be as above, but . Then there are factors that are all equal, that is, are not relatively prime. Hensel's lemma is not applicable. ${\displaystyle K,A,k}$ ${\displaystyle f(X)=X^{p}-1}$ ${\displaystyle {\bar {f}}(X)=X^{p}-{\bar {1}}=(X-{\bar {1}})^{p}}$ ${\displaystyle X-{\bar {1}}}$

Hensel's ring

→ Main article : Henselscher Ring

The assumption that is complete is actually stronger than it would be necessary for the proof of Hensel's lemma. In general, valued solids or rings in which Hensel's lemma applies in the form given above are called Henselian. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle A}$

Lifting tree

A raising tree is an aid to the behavior of a polynomial , or more precisely the behavior of the zeros modulo describe the polynomial. With the help of a lifting tree, one can more easily examine p-adic numbers and thus infer the behavior of the polynomial. ${\displaystyle f(X)\in \mathbb {Z} [X]}$ ${\displaystyle p^{k}}$

The elevation tree has the zeros modulo in its k-th level and these are connected with their elevations modulo if these are also zeros again. ${\displaystyle p^{k}}$ ${\displaystyle p^{k+1}}$

Zeros and their lifts

Let be a rationally irreducible polynomial . Let p be prime. Be the level of the elevation tree. ${\displaystyle f(X)\in \mathbb {Z} [X]}$ ${\displaystyle k\geq 1}$

Be . Is ${\displaystyle a\in \mathbb {Z} }$

${\displaystyle f(a)\equiv 0\mod p^{k}}$,

so we say, is a zero of f (X) in or modulo . ${\displaystyle a}$ ${\displaystyle \mathbb {Z} /p^{k}}$ ${\displaystyle p^{k}}$

Let be a zero of modulo . Be . Is a zero of f (X) modulo and is ${\displaystyle a\in \mathbb {Z} }$ ${\displaystyle f(X)}$ ${\displaystyle p^{k}}$ ${\displaystyle l\geq 1}$ ${\displaystyle b\in \mathbb {Z} }$ ${\displaystyle p^{k+l}}$

${\displaystyle b\equiv a\mod p^{k}}$ ,

then we say that a zero is modulo that raises the zero a modulo . ${\displaystyle b}$ ${\displaystyle p^{k+l}}$ ${\displaystyle p^{k}}$

Description of the lifting tree

In a lifting tree, all zeros of a polynomial are entered in, with the respective level of the lifting tree being. ${\displaystyle \mathbb {Z} /p^{k}}$ ${\displaystyle k\in \mathbb {Z} }$

The first level of the tree is at the very bottom. As the tree grows, it grows from bottom to top and all zeros are entered in the respective level. ${\displaystyle k}$

All zeros of the polynomial are entered in the lowest and thus first level ( ) . The zeros take on values ​​in the interval . ${\displaystyle k=1}$ ${\displaystyle \mathbb {Z} /p}$ ${\displaystyle [0,p-1]}$

In the second level above ( ) all zeros of the polynomial are entered. The zeros of the polynomial take on values ​​in the interval . If such a zero point in the second level reduces to a zero point in the first level below , these two zero points are connected with a line. ${\displaystyle k=2}$ ${\displaystyle \mathbb {Z} /p^{2}}$ ${\displaystyle [0,p^{2}-1]}$ ${\displaystyle \mathbb {Z} /p}$

In the next higher level ( ), all zeros of the polynomial are entered in. The zeros of the polynomial take on values ​​in the interval . The following also applies here: If a zero in the third level is reduced to a zero in the second level below , these zeros are connected with a line. ${\displaystyle k=3}$ ${\displaystyle \mathbb {Z} /p^{3}}$ ${\displaystyle [0,p^{3}-1]}$ ${\displaystyle \mathbb {Z} /p^{2}}$

This applies to all following levels . ${\displaystyle k\in \mathbb {Z} }$

example

Be the polynomial

${\displaystyle f(x)=x^{4}-3x^{3}-3x^{2}+x-1}$

given. Be prime. ${\displaystyle p=5}$

We get the following lifting tree:

Example of the polynomial ${\displaystyle x^{4}-3x^{3}-3x^{2}+x-1=0}$

In the first level ( ) there are zeros 1 and 3 in the interval . In the second level ( ) the zeros 3, 8, 13, 18 and 23 are present in the interval . In the following third level ( ) we see the zeros 8, 33, 58, 83 and 108 in the interval . It applies to this polynomial that all zeros in the second level are reduced to the zero in the first level. They are each connected with a line. In short: The zero point from the first level is raised to the second level. ${\displaystyle k=1}$ ${\displaystyle [0,4]}$ ${\displaystyle k=2}$ ${\displaystyle [0,24]}$ ${\displaystyle k=3}$ ${\displaystyle [0,124]}$ ${\displaystyle a=3}$ ${\displaystyle a=3}$

The same for the third level.

literature

• Jürgen Neukirch : Algebraic number theory . Springer-Verlag, Berlin 1992, ISBN 3-540-54273-6 .
• David Eisenbud: Commutative algebra. (= Graduate Texts in Mathematics. 150). Springer-Verlag, Berlin / New York 1995, ISBN 3-540-94268-8 .
• Atilla Pethö: Algebraic Algorithms . Ed .: Michael Pohst. Vieweg, 1999, ISBN 3-528-06598-2 , pp. 187 .

• Michael Kaplan: Computer Algebra . Springer, 2005, ISBN 3-540-21379-1 , pp. 51 .

• K. Hensel: Theory of Algebraic Numbers . Teubner, Leipzig 1908.
• Helmut Koch: Number Theory - Algebraic Numbers and Functions. Vieweg, 1997.
• Matthias Künzer: Lifting zeros. University of Stuttgart, 2011.

Individual evidence

1. ^ David A. Cox : Why Eisenstein proved the Eisenstein Criterion and why Schönemann discovered it first. In: American Mathematical Monthly. Volume 118, 2011, pp. 3-21.