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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Nontrivial Galois module structure of cyclotomic fields
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by Marc Conrad and Daniel R. Replogle PDF
Math. Comp. 72 (2003), 891-899 Request permission


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
  • Email:
  • Daniel R. Replogle
  • Affiliation: Department of Mathematics and Computer Science, College of Saint Elizabeth, 2 Convent Road, Morristown, New Jersey 07960
  • Email:
  • Received by editor(s): November 6, 2000
  • Received by editor(s) in revised form: July 15, 2001
  • Published electronically: June 4, 2002
  • © Copyright 2002 American Mathematical Society
  • Journal: Math. Comp. 72 (2003), 891-899
  • MSC (2000): Primary 11R33, 11R29; Secondary 11R27, 11R18
  • DOI:
  • MathSciNet review: 1954973