Noether's problem for groups of order 32

Let K be any field, 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 h . x(g) = x(hg) for any g, h is an element of G. Denote by K (G) = K(x(g): g is an element of G)(G) the fixed field. Noether's problem asks, under what situations, the fixed field K(G) will be rational (= purely transcendental) over K. Theorem. Let G be a finite group of order 32 with exponent e. If char K = 2 or K is any field containing a primitive eth root of unity, then K(G) is rational over K. (c) 2008 Elsevier Inc. All rights reserved.