The center of the generic G-crossed product

Let G   be a finite group and let FF be a field of characteristic zero. In this paper we construct a generic G  -crossed product over FF using generic graded matrices. The center of this generic G  -crossed product, denoted by F(G)F(G), is then the invariant field of a suitable G   action on a field of rational functions in several indeterminates. The main goal of this paper is to study the extensions F(G)F given that FF contains enough roots of unity and determine how close they are to being purely transcendental.In particular we show that F(G)/FF(G)/F is a stably rational extension for G=C2×C2nG=C2×C2n where n   is odd and for G=〈σ,τ|σn=τ2m=e,τστ−1=σ−1〉 where gcd(n,2m)=1gcd(n,2m)=1. Furthermore, we prove that if H,KH,K are groups of coprime orders, then F(H×K)F(H×K) is stably rationally equivalent to the fraction field of F(H)⊗F(K)F(H)⊗F(K).