Noether's problem for four- and five-dimensional linear actions

Let G be a finite group, K be a field and G→GL(V) be a faithful representation where V is a finite-dimensional vector space over K. In this paper, we will consider the problem which asks whether the fixed field KG(V) is rational (i.e. purely transcendental) over K. This is a variant of Noether's problem. We will show the complete results of this problem in the case where G is a 2-group, K=Q and and 5. Our results imply the existence of a generic polynomial for the corresponding group.