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Mathematics of Computation

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Bounds for computing the tame kernel

Author: Richard P. Groenewegen
Journal: Math. Comp. 73 (2004), 1443-1458
MSC (2000): Primary 11R70; Secondary 11Y40, 19C20
Published electronically: July 29, 2003
MathSciNet review: 2047095
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Abstract: The tame kernel of the $K_2$ of a number field $F$ is the kernel of some explicit map $K_2F\to \bigoplus k_v^*$, where the product runs over all finite primes $v$ of $F$ and $k_v$ is the residue class field at $v$. When $S$ is a set of primes of $F$, containing the infinite ones, we can consider the $S$-unit group $U_S$ of $F$. Then $U_S\otimes U_S$ has a natural image in $K_2F$. The tame kernel is contained in this image if $S$ contains all finite primes of $F$ up to some bound. This is a theorem due to Bass and Tate. An explicit bound for imaginary quadratic fields was given by Browkin. In this article we give a bound, valid for any number field, that is smaller than Browkin's bound in the imaginary quadratic case and has better asymptotics. A simplified version of this bound says that we only have to include in $S$ all primes with norm up to  $4\vert\Delta\vert^{3/2}$, where $\Delta$ is the discriminant of $F$. Using this bound, one can find explicit generators for the tame kernel, and a ``long enough'' search would also yield all relations. Unfortunately, we have no explicit formula to describe what ``long enough'' means. However, using theorems from Keune, we can show that the tame kernel is computable.

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Additional Information

Richard P. Groenewegen
Affiliation: Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RA Leiden, The Netherlands
Address at time of publication: ABN AMRO, Gustav Mahlerlaan 10, HQ 1056, 1082 PP Amsterdam, The Netherlands

Keywords: $K$-theory, tame kernel, calculations, $S$-units
Received by editor(s): April 18, 2002
Received by editor(s) in revised form: December 6, 2002
Published electronically: July 29, 2003
Article copyright: © Copyright 2003 American Mathematical Society