19 - Finale
from Part two - Factorization
Published online by Cambridge University Press: 15 October 2009
Summary
Kronecker's Dream
In concluding this part I want to give a résumé of the state of the art of polynomial factorization over K[X], K a field:
The Berlekamp and Cantor–Zassenhaus Algorithms allow factorization if K is a finite field;
The Berlekamp–Hensel–Zassenhaus and Lenstra–Lenstra–Lovász factorization algorithms allows us to lift factorization over ℤp to one over ℤ and, by the Gauss Lemma, to one over ℚ, so that factorization is available over the prime fields.
Algebraic extensions K = F(α) are dealt with by the Kronecker Algorithm (Section 16.3) if F is infinite, and by Berlekamp otherwise,
while Hensel–Zassenhaus allows us to factorize multivariate polynomials in F[X1, …, Xn] if factorization over F is available, so that
the Gauss Lemma allows us to deal with transcendental extensions K = F(X)
so that factorization is available over every field explicitly given in Kronecker's Model.
Van der Waerden's Example
Within the development of computational techniques for polynomial ideal theory, started by the benchmark work by G. Herrmann, van der Waerden pointed to a fascinating limitation of the ability to build fields within Kronecker's Model.
- Type
- Chapter
- Information
- Solving Polynomial Equation Systems IThe Kronecker-Duval Philosophy, pp. 415 - 419Publisher: Cambridge University PressPrint publication year: 2003