Polynomial factorization algorithms over number fields

Factorization algorithms over Q[X] and Fp[X] are key tools of computational number theory. Many algorithms over number fields rely on the possibility of factoring polynomials in those fields. Because of the recent development of relative methods in computational number theory, see for example (Cohen et al. 1998, Daberkow and Pohst 1995), efficient generalizations of factorization algorithms to the relative case are necessary. The aim of this paper is to describe two new such algorithms. The first algorithm factors polynomials modulo a prime ideal of a number field. It generalizes the algorithm of Berlekamp over Fp. The second factors polynomials over a number field. Note that there already exist algorithms performing this task, see (Trager 1976, Weinberger and Rothschild 1976, Lenstra 1982, Geddes et al. 1992). In fact, the method described in this paper can be viewed as a combination of the methods of Lenstra (1982) and Weinberger and Rothschild (1976). Notations and definitions are given in the first section. The second section is devoted to the factorization modulo a prime ideal, and the third to the description of the factorization algorithm over a number field. In the last section we give an example, several applications and some comparison timings.