Nontrivial Galois Module Structure of Cyclotomic Fields, by Marc Conrad and Daniel R. Replogle

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 tha t $\Cal{O}_{L}$ is a free $\Cal{O}_{K}[G]$-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 nont rivial Galois module structure. However, the proof does not directly yield specific primes $l$ for a given algebraic number field $K.$ For $K$ any cyclotomi c field we find an explicit $l$ so that there is a tame degree $l$ extension $L/K$ with nontrivial Galois module structure.

Marc Conrad and Daniel R. Replogle <dreplogle@liza.st-elizabeth.edu,marc@pension-perisic.de>