Let $E/F$ be a Galois extension of number fields with $\Gamma =$ Gal$(E/F)$ and with property that the divisors of $(E:F)$ are non-ramified in $E/\mathbf{Q}$. We denote the ring of integers of $E$ by ${𝒪}_E$ and we study ${𝒪}_E$ as a $\mathbf{Z}\Gamma$-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...