Hensel's lemma

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

It has been demonstrated as early as 1846 before Hensel of Theodor Beautiful man.

Formulation

It is a complete, non- archimedean valued body with valuation ring and residue field. Is now a polynomial whose reduction is the product of two relatively prime polynomials, polynomials there are, such that the reduction of and or or is.

Examples

• With the hensel between Lemma, one can show that the body of p- adic numbers containing the roots of unity:

• Is the prime number, then there is a according to the above with.

• In the body of the p -adic number 0 is represented by a sum of squares of non-trivial. -1 Is thus represented by a sum of squares.

• There are as above, but. Then with factors, all, so are not the same prime. The henselsche lemma is not applicable.

Hensel shear ring

The condition that is complete is actually too strong. General called evaluated body or rings in which the henselsche lemma holds in the form indicated above, henselsch.