An Efficient Algorithm for Factoring Polynomials over Algebraic Extension Field

An efficient algorithm is presented for factoring polynomials over an algebraic extension field. The extension field is defined by a polynomial ring modulo a maximal ideal. If the ideal is given by its Gröbner basis, no extra Gröbner basis computation is needed for factoring a polynomial over the extension field. We will only use linear algebra to get a polynomial over the base field by a generic linear map, and this polynomial will be factorized over the base field. From these factors, the factorization of the polynomial over the extension field can be obtained. The algorithm has been implemented and the experiments show that our algorithm is very efficient.