18 - Zassenhaus
from Part two - Factorization
Published online by Cambridge University Press: 15 October 2009
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.
- Type
- Chapter
- Information
- Solving Polynomial Equation Systems IThe Kronecker-Duval Philosophy, pp. 380 - 414Publisher: Cambridge University PressPrint publication year: 2003