Noether’s Problem for Some p-Groups

Summary

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.

2000 Mathematics Subject Classification codes: 12F12, 13A50, 11R32, 14E08