Fixed fields of triangular matrix groups

Let K be any field and G be a finite subgroup of GLn(K). Then G acts on the rational function field K(x1,x2,…,xn) by K-automorphisms defined by σ xj=∑1 i naijxi if σ=(aij) G. Let be the fixed field. Miyata shows that K(x1,…,xn)G is rational (i.e. purely transcendental) over K provided that G consists of upper triangular matrices. We will show that, in this situation, a transcendence basis f1,…,fn for K(x1,…,xn)G can be choosen with each fi being a polynomial in K[x1,…,xn]. In fact, this theorem follows from a more general result.