Noether’s Problem for Some p-Groups

Abstract

Let K be any field and G be a finite group. Let G act on the rational function field K(x g : g ∈ G) by K-automorphisms defined by g · x h = x gh for any g, h ∈ G. Noether’s problem asks whether the fixed field K(G) = K(x g : g ∈ G) G is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order p n containing a cyclic subgroup of index p and K is any field containing a primitive p n−2 -th root of unity, then K(G) is rational over K. As a corollary, if G is a non-abelian p-group of order p 3 and K is a field containing a primitive p-th root of unity, then K(G) is rational.