Nontrivial Galois module structure of cyclotomic fields

Abstract

We say a tame Galois field extension L / K L/K with Galois group G G has trivial Galois module structure if the rings of integers have the property that O L \mathcal {O}_{L} is a free O K [ G ] \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 l so that for each there is a tame Galois field extension of degree l l so that L / K L/K has nontrivial Galois module structure. However, the proof does not directly yield specific primes l l for a given algebraic number field K . K. For K K any cyclotomic field we find an explicit l l so that there is a tame degree l l extension L / K L/K with nontrivial Galois module structure.