19 - Finale

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 ℤ_p 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, …, X_n] 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.