18 - Zassenhaus

Summary

The impractical nature of von Schubert's Algorithm for factorization over ℤ is evident even from the example I have presented. The absence of a ‘reasonable’ factorization algorithm for polynomials over the integers was one of the major weaknesses of Kronecker's Model.

Berlekamp's Algorithm mended this flaw: in fact in 1969 Zassenhaus suggested substituting von Schubert's Algorithm with an application of Berlekamp's Algorithm and a lemma by Hensel.

Hensel's Lemma gives an algorithm which allows us to ‘lift’ a factorization over D/p to one over D/pⁿ where D is a principal ideal domain and p ∈ D is irreducible.

Zassenhaus proposed computing a factorization of a polynomial f over D, based on a factorization algorithm over D/p, by the following approach:

factorize the image of f over D/p;

lift, via Hensel, this factorization to one over D/pⁿ for a ‘suitably’ large n – the ‘suitability’ of n is based on the ability to recover all the coefficients of the factors of f over D – and

obtain the factors over D, by combining the ones over and checking if they divide f.