Stable rationality of certain invariant fields

Let F be a field. For a finite group G, let F(G) be the purely transcendental extension of F with transcendency basis {xg: g G}. Let F(G)G denote the fixed field of F(G) under the action of G. Let w be a primitive (p−1)st root of 1, and let I be the ideal (p,w−a) in Z[w] where a is a primitive (p−1)st root of 1modp. We show that if G be the semi-direct product of a cyclic group of order p by a cyclic group of order prime to p, if I is principal, and if F contains a primitive Gth root of 1, then F(G)G is stably rational over F. It is not known whether the set of primes p for which I is principal is finite or infinite. We also show that if p is an odd prime and G is a non-abelian group of order p3, then F(G)G is stably rational over F provided that F contains a primitive Gth root of 1.