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

DOI:
https://doi.org/10.1090/S0025-5718-00-01302-8

Published electronically:
October 18, 2000

MathSciNet review:
1863005

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the locally free class group, that is the group of stable isomorphism classes of locally free -modules, where is the ring of algebraic integers in the number field and is a finite group. We show how to compute the Swan subgroup, , of when , a primitive -th root of unity, , where is an odd (rational) prime so that and 2 is inert in 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 a nontrivial divisor of These calculations give an alternative proof that the fields for =11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

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Additional 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

DOI:
https://doi.org/10.1090/S0025-5718-00-01302-8

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