On factor refinement in number fields

Let O be an order of an algebraic number field. It was shown by Ge that giv en a factorization of an O-ideal a into a product of O-ideals it is possibl e to compute in polynomial time an overorder O' of O and a gcd-free refinem ent of the input factorization; i.e., a factorization of aO' into a power p roduct of O'-ideals such that the bases of that power product are all inver tible and pairwise coprime and the extensions of the factors of the input f actorization are products of the bases of the output factorization. In this paper we prove that the order O' is the smallest overorder of O in which s uch a gcd-free refinement of the input factorization exists. We also introd uce a partial ordering on the gcd-free factorizations and prove that the fa ctorization which is computed by Ge's algorithm is the smallest gcd-free re finement of the input factorization with respect to this partial ordering.