On factor refinement in number fields

Let $\mathcal O$ be an order of an algebraic number field. It was shown by Ge that given a factorization of an $\mathcal O$-ideal $\mathfrak{a}$ into a product of $\mathcal O$-ideals it is possible to compute in polynomial time an overorder $\mathcal O'$ of $\mathcal O$ and a gcd-free refinement of the input factorization; i.e., a factorization of $\mathfrak{a}\mathcal O'$ into a power product of $\mathcal O'$-ideals such that the bases of that power product are all invertible and pairwise coprime and the extensions of the factors of the input factorization are products of the bases of the output factorization. In this paper we prove that the order $\mathcal O'$ is the smallest overorder of $\mathcal O$ in which such a gcd-free refinement of the input factorization exists. We also introduce a partial ordering on the gcd-free factorizations and prove that the factorization which is computed by Ge's algorithm is the smallest gcd-free refinement of the input factorization with respect to this partial ordering.