On the essential dimension of cyclic p-groups

Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r >= 1 be an integer. We compute the essential dimension of Z/p(r) Z over K (Theorem 4.1). In particular, i) We have ed(Q)(Z/8Z)=4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). ii) We have ed(Q)(Z/p(r) Z) >= p(r-1).