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
DOI:
https://doi.org/10.1090/S0025-5718-99-01023-6
MathSciNet review:
1613766
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Abstract | References | Similar Articles | Additional Information
Abstract: Let be an order of an algebraic number field. It was shown by Ge that given a factorization of an
-ideal
into a product of
-ideals it is possible to compute in polynomial time an overorder
of
and a gcd-free refinement of the input factorization; i.e., a factorization of
into a power product of
-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
is the smallest overorder of
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.
- [BDS93] E. Bach, J. Driscoll, and J. Shallit, Factor refinement, J. Algorithms 15 (1993), 199-222. MR 94m:11148
- [Ge93] Guoqiang Ge, Algorithms related to multiplicative representations of algebraic numbers, PhD thesis, U.C. Berkeley, 1993.
- [Ge94] Guoqiang Ge, Recognizing units in number fields, Math. Comp. 63 (1994), 377-387. MR 94i:11107
- [ZS58] O. Zariski and P. Samuel, Commutative algebra, Van Nostrand, Princeton, 1958. MR 19:833e
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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
DOI:
https://doi.org/10.1090/S0025-5718-99-01023-6
Received by editor(s):
November 21, 1996
Article copyright:
© Copyright 1999
American Mathematical Society