On the projective class group of cyclic groups of prime power order

Abstract

Let Cq denote the cyclic group of order q and ZCq the integral group ring of Cq. If q is a prime, q = p say, D. S. Rim [18] has proved that the projective class group ITio(ZCp) is isomorphic to /(o(Z[~]), where ~ denotes a primitive p-th root of unity. In turn, it is well known that/(o(Z[~]) is isomorphic to the ideal class group of the ring Z[~] of integers in the cyclotomic field Fo = Q(~). See J. Milnor's book [17], w Corollary 1.11. In this paper we study fflo(ZCq) for q = p,+l, where p is a prime number. For instance, we obtain in w the following result. Let C(Fn) denote the ideal class group of the cyclotomic field F, = Q(~,), .where ~, is a primitive pn+l-st root of unity. If p is a semi-regular odd prime, there is an exact sequence