Journal of the Mathematical Society of Japan

Some cases of four dimensional linear Noether's problem


Full-text: Open access


The linear Noether's problem means the rationality problem for the fixed field of linear actions on the rational function field. This paper deals with a part of our study on the four dimensional linear Noether's problem. Apart from the main part of our study, which will be published in other papers, the results which require complicated calculations by a computer are published here as a separate paper. The problem is affirmative for all of 5 non-solvable subgroups and the largest and one of the second largest subgroups of GL(4,Q). As relevant topics, we remark that PSp(3,2) (the simple group of order 1451520) has a generic polynomial over Q.

Article information

J. Math. Soc. Japan Volume 62, Number 4 (2010), 1273-1288.

First available in Project Euclid: 2 November 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 12F20: Transcendental extensions
Secondary: 12F12: Inverse Galois theory

linear Noether's problem finite reflection groups generic polynomial inverse Galois theorem


YAMASAKI, Aiichi. Some cases of four dimensional linear Noether's problem. J. Math. Soc. Japan 62 (2010), no. 4, 1273--1288. doi:10.2969/jmsj/06241273.

Export citation


  • H. Brown, R. Bulow, J. Neubuser, H. Wondratschek and H. Zassenhaus, Crystallographic groups of four-dimensional space, John Wiley, New York, 1978.
  • F. R. DeMeyer, Generic polynomials, J. Algebra, 84 (1983), 441–448.
  • L. C. Grove and C. T. Benson, Finite Reflection Groups, 2nd ed., Springer-Verlag, 1985.
  • C. Jensen, A. Ledet and N. Yui, Generic polynomials, constructive aspects of the inverse Galois problem, Cambridge, 2003.
  • M. Kang, Noether's problem for dihedral 2-groups II, Pacific J. Math., 222 (2005), 301–316.
  • G. Kemper and E. Mattig, Generic polynomials with few parameters, J. Symbolic. Comput., 30 (2000), 843–857.
  • G. Kemper, Generic polynomials are descent-generic, Manuscripta Math., 105 (2001), 139–141.
  • H. Kitayama, Linear Noether's problem for 2-groups, J. Algebra, 324 (2010), 591–597.
  • H. Kitayama and A. Yamasaki, Four-dimensional linear Noether's problem, J. Math. Kyoto Univ., 49 (2009), 359–380.
  • H. W. Lenstra, Jr., Rational functions invariant under a finite abelian group, Invent. Math., 25 (1974), 299–325.
  • T. Maeda, Noether's problem for $\mathfrak{A}_5$, J. Algebra, 125 (1989), 418–430.
  • J. F. Mestre, Correspondances compatibles avec une relation binaire, relèvement d'extensions de groupe de Galois $L_3(2)$ et problème de Noether pour $L_3(2)$,, 2005.
  • B. Plans, Noether's problem for $GL(2,3)$, Manuscripta Math., 124 (2007), 481–487.
  • Y. Rikuna, The existence of generic polynomial for $SL(2,3)$ over $\bm{Q}$, preprint.