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Let $G$ be a finite group and let $N/E$ be a tamely ramified $G$-Galoisextension of number fields. We show how Stickelberger's factorization of Gausssums can be used to determine the stable isomorphism class of variousarithmetic $\mathbb{Z}[G]$-modules attached to $N/E$. If $\mathcal{O}_N$ and$\mathcal{O}_E$ denote the rings of integers of $N$ and $E$ respectively, weget in particular that $\mathcal{O}_N\otimes_{\mathcal{O}_E}\mathcal{O}_N$defines the trivial class in the class group $\mathrm{Cl}(\mathbb{Z}[G])$ and,if $N/E$ is also assumed to be locally abelian, that the square root of theinverse different (whenever it exists) defines the same class as$\mathcal{O}_N$. These results are obtained through the study of the Fr\"ohlichrepresentatives of the classes of some torsion modules, which are independentlyintroduced in the setting of cyclotomic number fields. Gauss and Jacobi sums,together with the Hasse-Davenport formula, are involved in this study. Thesetechniques are also applied to recover the stable self-duality of$\mathcal{O}_N$ (as a $\mathbb{Z}[G]$-module). Finally, when $G$ is the binarytetrahedral group, we use our results in conjunction with Taylor's theorem tofind a tame $G$-Galois extension whose square root of the inverse different hasnontrivial class in $\mathrm{Cl}(\mathbb{Z}[G])$.

Gauss sums, Jacobi sums and cyclotomic units related to torsion Galois modules