Norms and traces in p-adic Abelian group rings
The paper under review is to be seen as a further step in the direction of describing integral group ring units in the abelian case in terms of those of the cyclic subgroups. Denote by \(U'_*A\) the subgroup of units in \((1+\Delta_*A)'\) where \(\Delta\) (A) is the augmentation ideal in the p-adic group ring \({\mathbb{Z}}_ pA\) of an abelian p-group A; the * indicates that we only worry about elements that are invariant under \(a\mapsto a^{-1}\) (a\(\in A)\), the ' indicates that we restrict ourselves to looking at the elements having G-norm 1 with G being the automorphism group of \({\mathbb{Z}}/p^ m\) where \(p^ m=\exp A\). Theorem: \(U'_*(A)\) is generated by all \(U'_*(C)\), C running through the cyclic subgroups of A, and \(U'_*(A)\to U'_*(A/B)\) is surjective for any \(B\leq A\). The paper pursues work started already by the author and the reviewer [in Arch. Math. 49, 23-28 (1987; Zbl 0649.16011)] and the author and \textit{S. K. Sehgal} [in J. Number Theory 30, No.3, 375-381 (1988; Zbl 0654.16006)].
scientific article; zbMATH DE number 4061440