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Nontrivial Galois module structure of cyclotomic fields

Authors: Marc Conrad and Daniel R. Replogle
Journal: Math. Comp. 72 (2003), 891-899
MSC (2000): Primary 11R33, 11R29; Secondary 11R27, 11R18
Published electronically: June 4, 2002
MathSciNet review: 1954973
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Abstract: We say a tame Galois field extension $L/K$ with Galois group $G$ has trivial Galois module structure if the rings of integers have the property that $\mathcal{O}_{L}$ is a free $\mathcal{O}_{K}[G]$-module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes $l$ so that for each there is a tame Galois field extension of degree $l$ so that $L/K$ has nontrivial Galois module structure. However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For $K$ any cyclotomic field we find an explicit $l$ so that there is a tame degree $l$extension $L/K$ with nontrivial Galois module structure.

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Additional Information

Marc Conrad
Affiliation: Faculty of Technology, Southampton Institute, East Park Terrace, Southampton, S014 0YN Great Britain

Daniel R. Replogle
Affiliation: Department of Mathematics and Computer Science, College of Saint Elizabeth, 2 Convent Road, Morristown, New Jersey 07960

Keywords: Swan subgroups, cyclotomic units, Galois module structure, tame extension, normal integral basis
Received by editor(s): November 6, 2000
Received by editor(s) in revised form: July 15, 2001
Published electronically: June 4, 2002
Article copyright: © Copyright 2002 American Mathematical Society