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

✍ Scribed by Alexander Stolin

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
324 KB
Volume
72
Category
Article
ISSN
0022-314X

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✦ Synopsis

In the present paper we deal with the canonical projection Pic Z

Here p is any odd prime number, pk k =1 and C n is the cyclic group of order p n . I proved in (Stolin, 1997), that the canonical projection Pic Z[n] Γ„ Cl Z[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 ] Γ„ Cl Z[n &1 ] does not split and the p-component of Cl Z[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(1).

1998 Academic Press

A k, i =Z[x k, i ] <\ x p k+i k, i &1 x p k k, i &1 + i 1, k 0.

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