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

Abstract

In the present paper we deal with the canonical projection Pic Z [ C n ]→⊕ n k=0 Cl Z [ ζ k ]. Here pis any odd prime number, ζ p k k =1 and C n is the cyclic group of order p n . I proved in (Stolin, 1997), that the canonical projection Pic Z [ ζ n ]→Cl Z [ ζ n ] can be split. If pis 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).