Let K be any field, G be a finite group. Let G act on the rational function field by K-automorphisms defined by h⋅xg=xhg for any g,h∈G. Denote by the fixed field. Noether's problem asks, under what situations, the fixed field K(G) will be rational (= purely transcendental) over K.
. Let G be a finite group of order 32 with exponent e. IfcharK=2or K is any field containing a primitive eth root of unity, thenK(G)is rational over K.