Fixed fields of triangular matrix groups
Journal
Journal of Algebra
Journal Volume
302
Journal Issue
2
Pages
845-847
Date Issued
2006
Author(s)
Abstract
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 ṡ xj = ∑1 ≤ i ≤ n ai j xi if σ = (ai j) ∈ G. Let K (x1, ..., xn)G = {f ∈ K (x1, ..., xn) : σ ṡ f = f for any σ ∈ G} 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. © 2006 Elsevier Inc. All rights reserved.
Type
journal article