The ring of integers of an Abelian extension of an algebraic number field as a Galois module

Abstract

The ringO of integers of a finite Abelian extension K of an algebraic number field k is studied as a module over the group ring Λ=σ[G], where σ is the ring of integers of k and G is the Galois group of K/k. It is proved that the ring σ is a decomposable Λ-module if and only if there exists in K/k an intermediate extension K/F. F≠K, whose degree divides the different.