## Computation of several cyclotomic Swan subgroups

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- by Timothy Kohl and Daniel R. Replogle;
- Math. Comp.
**71**(2002), 343-348 - DOI: https://doi.org/10.1090/S0025-5718-00-01302-8
- Published electronically: October 18, 2000
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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

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## Bibliographic Information

**Timothy Kohl**- Affiliation: Office of Information Technology, Boston University, Boston, Massachusetts
- Email: tkohl@math.bu.edu
**Daniel R. Replogle**- Affiliation: Department of Mathematics and Computer Science, College of Saint Elizabeth, Morristown, New Jersey
- Email: dreplogle@liza.st-elizabeth.edu
- Received by editor(s): August 14, 1998
- Received by editor(s) in revised form: March 1, 2000
- Published electronically: October 18, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp.
**71**(2002), 343-348 - MSC (2000): Primary 11R33, 11R18
- DOI: https://doi.org/10.1090/S0025-5718-00-01302-8
- MathSciNet review: 1863005