Noether's problem for metacyclic p-groups

Let K be any field and G be a finite group. Let G act on the rational function field K(x(g) : g is an element of G) by K-automorphisms defined by g . x(h) = x(gh) for any g, h is an element of G. Denote by K(G) the fixed field K(x(g) : g is an element of G)(G). Noether's problem asks whether K(G) is rational (=purely transcendental) over K. An affirmative answer to Noether's problem for metacyclic p-groups will be proved provided that K contains enough roots of unity. (C) 2005 Elsevier Inc. All rights reserved.