Noether΄s Problem for Some p-Groups.

Let K be any field and G be a finite group. Let G act on the rational function field K(x_g : gϵG) by K-automorphisms defined by g · x_h = x_gh for any g, hϵG. Noether΄s problem asks whether the fixed field K(G) = K(x_g : gϵG)^G is rational (=purely transcendental) over K. We will prove that if G is a non-abelian p-group of order pⁿ containing a cyclic subgroup of index p and K is any field containing a primitive pⁿ⁻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 p³ and K is a field containing a primitive p-th root of unity, then K(G) is rational. [ABSTRACT FROM AUTHOR]