On factor refinement in number fields
HTML articles powered by AMS MathViewer
- by Johannes Buchmann and Friedrich Eisenbrand PDF
- Math. Comp. 68 (1999), 345-350 Request permission
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.References
- Eric Bach, James Driscoll, and Jeffrey Shallit, Factor refinement, J. Algorithms 15 (1993), no. 2, 199–222. MR 1231441, DOI 10.1006/jagm.1993.1038
- Guoqiang Ge, Algorithms related to multiplicative representations of algebraic numbers, PhD thesis, U.C. Berkeley, 1993.
- Guoqiang Ge, Recognizing units in number fields, Math. Comp. 63 (1994), no. 207, 377–387. MR 1242057, DOI 10.1090/S0025-5718-1994-1242057-X
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
Additional Information
- Johannes Buchmann
- Affiliation: Technische Hochschule Darmstadt, Alexanderstr. 10, D-64283 Darmstadt, Germany
- Email: buchmann@cdc.informatik.th-darmstadt.de
- Friedrich Eisenbrand
- Affiliation: Max-Planck-Institut für Informatik, Im Stadtwald, D-66123 Saarbrücken, Germany
- Email: eisen@mpi-sb.mpg.de
- Received by editor(s): November 21, 1996
- © Copyright 1999 American Mathematical Society
- Journal: Math. Comp. 68 (1999), 345-350
- MSC (1991): Primary 11Y40, 11R27, 11R04, 11Y16
- DOI: https://doi.org/10.1090/S0025-5718-99-01023-6
- MathSciNet review: 1613766