An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field

TL;DR

This paper introduces a new efficient algorithm for factoring polynomials over algebraic extension fields, utilizing linear algebra techniques and Groebner basis without additional Groebner basis computations.

Contribution

The paper presents a novel algorithm that simplifies polynomial factorization over extension fields by avoiding extra Groebner basis calculations and employing linear algebra methods.

Findings

Algorithm is highly efficient in complex examples

No extra Groebner basis computation needed

Implemented algorithm shows promising results

Abstract

A new efficient algorithm is proposed for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the maximal ideal is given by its Groebner basis, no extra Groebner basis computation is needed for factoring a polynomial over this extension field. Nothing more than linear algebraic technique is used to get a polynomial over the ground field by a generic linear map. Then this polynomial is factorized over the ground field. From these factors, the factorization of the polynomial over the extension field is obtained. The new algorithm has been implemented and computer experiments indicate that the new algorithm is very efficient, particularly in complicated examples.