LMS Journal of Computation and Mathematics

Research Article

Computations in Relative Algebraic K-Groups

Werner Bleya1 and Stephen M. J. Wilsona2

a1 Fachbereich für Mathematik der Universität Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germany, bley@mathematik.uni-kassel.de

a2 Heilbronn Institute for Mathematical Research, University Of Bristol, Royal Fort Annexe, Clifton, Bristol BS8 1TW, United Kingdom, s.m.j.wilson@durham.ac.uk

Abstract

Let G be finite group and K a number field or a p-adic field with ring of integers OK. In the first part of the manuscript we present an algorithm that computes the relative algebraic K-group K0(OK[G], K) as an abstract abelian group. We also give algorithms to solve the discrete logarithm problems in K0(OK[G], K) and in the locally free class group cl(OK[G]). All algorithms have been implemented in Magma for the case K = Q.

In the second part of the manuscript we prove formulae for the torsion subgroup of K0(Z[G], Q) for large classes of dihedral and quaternion groups.

(Received February 28 2008)

(Revised March 17 2009)