Galois module structure of rings of integers

Abstract

Let E/F be a Galois extension of number fields with Γ= Gal(E/F) and with property that the divisors of (E:F) are non-ramified in E/Q. We denote the ring of integers of E by 𝒪 E and we study 𝒪 E as a ZΓ-module. In particular we show that the fourth power of the (locally free) class of 𝒪 E is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of ℰ E , together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.