STABLE RATIONALITY OF CERTAIN INVARIANT FIELDS

Esther Beneish
Eastern Washington University
Department of Mathematics
Cheney,  WA 99004

Abstract:
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  belongs to G}.  Let F(G)^G be the corresponding fixed field. 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-1st root of 1 mod p.  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 pth 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 p^3, then F(G)^G is stably rational over F provided that F contains a primitive p^2-th root of 1.

Status: Preprint.