Some cases of four dimensional linear Noether's problem
type: Item
instance of: scholarly article
title: Some cases of four dimensional linear Noether's problem
zbMATH Open document ID: 1219.14019
author: Aiichi Yamasaki
published in: Journal of the Mathematical Society of Japan
publication date: 2011-01-12T00:00:00Z
review text: This paper is a continuation of the paper by \textit{H. Kitayama} and \textit{A. Yamasaki} [J. Math. Kyoto Univ. 49, No. 2, 359--380 (2009; Zbl 1188.13004)]. The reader is referred to the review in Zentralblatt Math for a short account of the rationality problem discussed in Kitayama-Yamasaki's paper and this paper. In the paper of Kitayama and Yamasaki, several cases were left unproved: (i) the groups whose GAP codes are \((4,33,14)\), \((4,33,16)\) and \((4,31,i)\) where \(3\leq i\leq7\), and (ii) the groups \((4,33,3)\), \((4,33,6)\), \((4,33,7)\) and \((4,33,11)\). The authors promised to publish the proof for groups in case (i) in a separate paper; this paper is just the one promised before. The question for groups in case (ii) was solved recently by the reviewer and \textit{J. Zhou} [``The rationality problem for finite subgroups of \(\text{GL}_4(\mathbb Q)\)'', \url{arXiv:1006.1156} (2010)] as shown in the note added in proof of this paper under review. The proof for groups of \((4,31,i)\) where \(3\leq i\leq7\) uses quite a lot of computer computation. For an alternative proof, the reader may find another proof in the paper of the reviewer and J. Zhou mentioned before. The proof for the groups \((4,33,14)\) and \((4,33,16)\) is contained in Section 3. The reader is warned of a mis-numbering in the paper of Kitayama and Yamasaki (p. 378). There the reader was told that the rationality for the group \((4,33,14)\) had been given while the proof for the groups of \((4,33,15)\) and \((4,33,16)\) would be delayed to this paper. This is not true. The fact is what had been solved in Kitayama and Yamasaki's paper was the group \((4,33,15)\) instead of the group \((4,33,14)\). In the Appendix of this paper, the author remarks that the simple group \(G = \text{PSp}(3, 2)\) has a generic polynomial over the rational number field. The proof goes as follows. The group \(G \times C_2\) is isomorphic to the finite reflection group \(E_7\). The ring of invariants associated …
reviewed by: Ming-Chang Kang
Mathematics Subject Classification ID: 14E08
Mathematics Subject Classification ID: 12F20
Mathematics Subject Classification ID: 12F12
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 5835143
zbMATH Keywords: rationality problem
zbMATH Keywords: Noether's problem
zbMATH Keywords: inverse Galois problem
zbMATH Keywords: finite reflection groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/06241273
OpenAlex ID: W2149754783
cites work: Q4161848
cites work: Generic polynomials
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DOI: 10.2969/JMSJ/06241273
description: scientific article; zbMATH DE number 5835143