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

In the present paper we deal with the canonical projection Pic image[Cn]→circled plusnk=0 Cl image[ζk]. Herepis any odd prime number,ζpkk=1 andCnis the cyclic group of orderpn. I proved in (Stolin, 1997), that the canonical projection Pic image[ζn]→Cl image[ζn] can be split. Ifpis a properly irregular, not regular prime number, then we prove in this paper that the projection Pic image[Cn]→Cl image[ζn−1] does not split and thep-component of Cl image[ζn−1] is an obstruction for the splitting. We construct an embedding of the Tate moduleTp(image) into Pic (proj.limit image[Cn]). Using an exact formula for Pic image[C2] we obtain a formula for the Galois group of a certain extension of image(ζ1).