Hensel’s lemma for integers
The solution of (1) may be refined in its residue class modulo to a solution of the congruence
This refinement is unique modulo iff .
Proof. Now we have . We have to find an such that
The short Taylor theorem requires that
where , whence this congruence can be simplified to
Thus the integer must satisfy the linear congruence
When , this congruence has a unique solution modulo (see linear congruence); thus we have the refinement which is unique modulo .
When and , the congruence evidently is impossible.
In the case the congruence (2) is identically true in the residue class of modulo . □
|Title||Hensel’s lemma for integers|
|Date of creation||2013-04-08 19:35:26|
|Last modified on||2013-04-08 19:35:26|
|Last modified by||pahio (2872)|