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

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Let G be a finite group and let N/E be a tamely ramified G-Galois extension of number fields. We show how Stickelberger's factorization of Gauss sums can be used to determine the stable isomorphism class of various arithmetic Z[G]-modules attached to N/E. If Oₙ and Oₑ denote the rings of integers of N and E respectively, we get in particular that Oₙ⊗ₒₑOₙ defines the trivial class in the class group Cl(Z[G]) and, if N/E is also assumed to be locally abelian, that the square root of the inverse different (whenever it exists) defines the same class as Oₙ. These results are obtained through the study of the Fröhlich representatives of the classes of some torsion modules, which are independently introduced in the setting of cyclotomic number fields. Gauss and Jacobi sums, together with the Hasse-Davenport formula, are involved in this study. These techniques are also applied to recover the stable self-duality of Oₙ (as a Z[G]-module). Finally, when G is the binary tetrahedral group, we use our results in conjunction with Taylor's theorem to find a tame G-Galois extension whose square root of the inverse different has nontrivial class in Cl(Z[G]).