Nonrational fixed fields (Q912152)

Let L/K and K/F be finitely generated field extensions. If \(K=F(X_ 1,...,X_ n)\) where \(\{X_ 1,...,X_ n\}\) is algebraically independent over F, then K is a rational extension of F. Saltman defined K to be a retract rational extension of F if K is the quotient field of an F- algebra A and there are maps f: F[X\({}_ 1,...,X_ n](1/W)\to A\) and g: \(A\to F[X_ 1,...,X_ n](1/W)\) such that \(f\circ g=id\), where \(\{X_ 1,...,X_ n\}\) is algebraically independent over F and \(W\in F[X_ 1,...,X_ n].\) Let G be a finite group of K-automorphisms of a rational function field \(k(X_ 1,...,X_ n)\). Assume that the function fields \(k(X_ 1,...,X_ i)\), \(i=1,...,n\), are stabilized by G. Triantaphyllou showed that if \(| G|\) is odd, then the fixed field of G will be rational over k. The authors give an example where the order of G is 2 and the fixed field of G is not even retract rational.