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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Computing the multiplicative group of residue class rings


Authors: Florian Heß, Sebastian Pauli and Michael E. Pohst
Journal: Math. Comp. 72 (2003), 1531-1548
MSC (2000): Primary 11R29, 11R37, 11Y16, 11Y40
Published electronically: January 13, 2003
MathSciNet review: 1972751
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Abstract: Let $\mathbf{k}$ be a global field with maximal order $\mathfrak o_{\mathbf k}$and let ${\mathfrak{m}}_{0}$ be an ideal of $\mathfrak o_{\mathbf k}$. We present algorithms for the computation of the multiplicative group $(\mathfrak o_{\mathbf k}/{\mathfrak{m}}_{0})^*$ of the residue class ring $\mathfrak o_{\mathbf k}/{\mathfrak{m}}_{0}$ and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group $\mathbf{Cl}_{\mathbf{k}}^{\mathfrak{m}}$ modulo $\mathfrak m={\mathfrak{m}}_{0}{\mathfrak{m}}_{\infty}$, where ${\mathfrak{m}}_{\infty}$ denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.


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Additional Information

Florian Heß
Affiliation: Institut für Mathematik, MA 8–1, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
Address at time of publication: Department of Computer Science, University of Bristol, BS8 1UB, England
Email: florian@cs.bris.ac.uk

Sebastian Pauli
Affiliation: Institut für Mathematik, MA 8–1, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
Email: pauli@math.tu-berlin.de

Michael E. Pohst
Affiliation: Institut für Mathematik, MA 8–1, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany
Email: pohst@math.tu-berlin.de

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01474-1
PII: S 0025-5718(03)01474-1
Received by editor(s): February 2, 1999
Received by editor(s) in revised form: November 8, 2001
Published electronically: January 13, 2003
Article copyright: © Copyright 2003 American Mathematical Society