On the Picard Group of the Integer Group Ring of the Cyclic p-Group and Certain Galois Groups

摘要

In the present paper we deal with the canonical projection Pic Z[C_n]→?_(k = 0)~n Cl Z[ζ_k]. Here p is any odd prime number, ζ_k~(p~k) = 1 and C_n is the cyclic group of order p~n. I proved in (Stolin, 1997), that the canonical projection Pic Z[250L芲n]→Cl Z[ζ_n] can be split. If p is a properly irregular, not regular prime number, then we prove in this paper that the projection Pic Z[C_n]→Cl Z[ζ_(n-1)] does not split and the p-component of Cl Z[ζ_(n-1)] is an obstruction for the splitting. We construct an embedding of the Tate module T_p(Q) into Pic (proj.limit Z[C_n]). Using an exact formula for Pic Z[C_2] we obtain a formula for the Galois group of a certain extension of Q(ζ_1).