Fixed fields of triangular matrix groups
type: Item
instance of: scholarly article
title: Fixed fields of triangular matrix groups
zbMATH Open document ID: 1104.13002
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: Let \(K\) be a field, \(G\) a finite subgroup of \(\text{GL}_n(K)\) acting on the rational function field \(K(x_1,\ldots, x_n)\) by \(K\)-automorphism in the usual manner; let \(K(x_1,\ldots,x_n)^G\) denote the subfield of functions invariant under the action of \(G.\) It was shown by \textit{T. Miyata} [Nagoya Math. J. 41, 69--73 (1971; Zbl 0211.06801)] that if \(G\) consists of upper triangular matrices, then \(K(x_1,\ldots,x_n)^G\) is a purely transcendental extension of \(K.\) In the present note, using more general results of \textit{H. Ahmad, M. Hajja} and \textit{M. Kang} [J. Algebra 228, 643--658 (2000; Zbl 0993.12003)], it is shown that there actually exists a transcendence basis \(f_1,\ldots, f_n \in K[x_1,\ldots, x_n],\) so that \(K(f_1,\ldots, f_n)=K(x_1,\ldots,x_n)^G.\) Note: The references should give \textit{T. A. Springer}'s book [``Invariant theory'' (1977; Zbl 0346.20020)] as LNM 585 (not as GTM 585).
reviewed by: Alexander Kovačec
Mathematics Subject Classification ID: 13A50
zbMATH DE Number: 5078151
zbMATH Keywords: Noether's problem
zbMATH Keywords: rationality problem
zbMATH Keywords: triangular matrix groups
zbMATH Keywords: polynomial invariants
author: Ming-Chang Kang
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.03.037
OpenAlex ID: W2071099470
cites work: Rationality of some projective linear actions
cites work: On the Fixed Field of a Linear Abelian Group
cites work: Finite group actions on rational function fields
cites work: Invariants of Certain Groups I
cites work: Invariant theory
DOI: 10.1016/J.JALGEBRA.2006.03.037
description: scientific article; zbMATH DE number 5078151