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Mathematics of Computation

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Computation of several cyclotomic Swan subgroups

Authors: Timothy Kohl and Daniel R. Replogle
Journal: Math. Comp. 71 (2002), 343-348
MSC (2000): Primary 11R33, 11R18
Published electronically: October 18, 2000
MathSciNet review: 1863005
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Abstract: Let $Cl(\mathcal{O}_{K}[G])$ denote the locally free class group, that is the group of stable isomorphism classes of locally free $\mathcal{O}_{K}[G]$-modules, where $\mathcal{O}_{K}$ is the ring of algebraic integers in the number field $K$ and $G$ is a finite group. We show how to compute the Swan subgroup, $T(\mathcal{O}_{K}[G])$, of $Cl(\mathcal{O}_{K}[G])$ when $K=\mathbb{Q} (\zeta_{p})$, $\zeta_{p}$ a primitive $p$-th root of unity, $G=C_{2}$, where $p$ is an odd (rational) prime so that $h_{p}^{+}=1$ and 2 is inert in $K/\mathbb{Q} .$ We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ring; we do the computations obtaining for several primes $p$ a nontrivial divisor of $Cl(\mathbb{Z} [\zeta_{p}]C_{2}).$ These calculations give an alternative proof that the fields $\mathbb{Q} (\zeta_{p})$ for $p$=11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

References [Enhancements On Off] (What's this?)

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Additional Information

Timothy Kohl
Affiliation: Office of Information Technology, Boston University, Boston, Massachusetts

Daniel R. Replogle
Affiliation: Department of Mathematics and Computer Science, College of Saint Elizabeth, Morristown, New Jersey

Received by editor(s): August 14, 1998
Received by editor(s) in revised form: March 1, 2000
Published electronically: October 18, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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