The center of the generic G-crossed product

Abstract

Let G be a finite group and let F be a field of characteristic zero. In this paper we construct a generic G-crossed product over F using generic graded matrices. The center of this generic G-crossed product, denoted by 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 F contains enough roots of unity and determine how close they are to being purely transcendental. In particular we show that F(G)/F is a stably rational extension for $G = C_2 \times C_{2n}$ where n is odd and for $G=<{\sigma},{\tau} | {\sigma}^n = {\tau}^{2m} = e, {\tau}{\sigma}{\tau}^{-1}={\sigma}^{-1}>$ where $gcd(n, 2m) = 1$. Furthermore, we prove that if H, K are groups of coprime orders, then $F(H \times K)$ is the fraction field of $F(H) \otimes F(K)$.