Nontrivial Galois module structure of cyclotomic fields

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.