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Mathematics of Computation

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On factor refinement in number fields

Authors: Johannes Buchmann and Friedrich Eisenbrand
Journal: Math. Comp. 68 (1999), 345-350
MSC (1991): Primary 11Y40, 11R27, 11R04, 11Y16
MathSciNet review: 1613766
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Abstract: 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.

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Additional Information

Johannes Buchmann
Affiliation: Technische Hochschule Darmstadt, Alexanderstr. 10, D-64283 Darmstadt, Germany

Friedrich Eisenbrand
Affiliation: Max-Planck-Institut für Informatik, Im Stadtwald, D-66123 Saarbrücken, Germany

Received by editor(s): November 21, 1996
Article copyright: © Copyright 1999 American Mathematical Society