Some cases of four dimensional linear Noether's problem
Aiichi YAMASAKI
Source: J. Math. Soc. Japan Volume 62, Number 4
(2010), 1273-1288.
Abstract
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.
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Keywords: linear Noether's problem; finite reflection groups; generic polynomial; inverse Galois theorem
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jmsj/1288703105
Digital Object Identifier: doi:10.2969/jmsj/06241273
Zentralblatt MATH identifier: 05835143
Mathematical Reviews number (MathSciNet): MR2761897
References
H. Brown, R. Bulow, J. Neubuser, H. Wondratschek and H. Zassenhaus, Crystallographic groups of four-dimensional space, John Wiley, New York, 1978.
Mathematical Reviews (MathSciNet): MR484179
Zentralblatt MATH: 0381.20002
F. R. DeMeyer, Generic polynomials, J. Algebra, 84 (1983), 441–448.
Mathematical Reviews (MathSciNet): MR723401
Digital Object Identifier: doi:10.1016/0021-8693(83)90087-X
L. C. Grove and C. T. Benson, Finite Reflection Groups, 2nd ed., Springer-Verlag, 1985.
Mathematical Reviews (MathSciNet): MR777684
Zentralblatt MATH: 0579.20045
C. Jensen, A. Ledet and N. Yui, Generic polynomials, constructive aspects of the inverse Galois problem, Cambridge, 2003.
Mathematical Reviews (MathSciNet): MR1969648
Zentralblatt MATH: 1042.12001
M. Kang, Noether's problem for dihedral 2-groups II, Pacific J. Math., 222 (2005), 301–316.
Mathematical Reviews (MathSciNet): MR2225074
Zentralblatt MATH: 1100.12003
Digital Object Identifier: doi:10.2140/pjm.2005.222.301
G. Kemper and E. Mattig, Generic polynomials with few parameters, J. Symbolic. Comput., 30 (2000), 843–857.
Mathematical Reviews (MathSciNet): MR1800681
Zentralblatt MATH: 0974.12004
Digital Object Identifier: doi:10.1006/jsco.1999.0385
G. Kemper, Generic polynomials are descent-generic, Manuscripta Math., 105 (2001), 139–141.
Mathematical Reviews (MathSciNet): MR1885819
Zentralblatt MATH: 1023.12003
Digital Object Identifier: doi:10.1007/s002290170015
H. Kitayama, Linear Noether's problem for 2-groups, J. Algebra, 324 (2010), 591–597.
Mathematical Reviews (MathSciNet): MR2651559
Zentralblatt MATH: 1200.13012
Digital Object Identifier: doi:10.1016/j.jalgebra.2010.04.029
H. Kitayama and A. Yamasaki, Four-dimensional linear Noether's problem, J. Math. Kyoto Univ., 49 (2009), 359–380.
Mathematical Reviews (MathSciNet): MR2571847
Zentralblatt MATH: 1188.13004
Project Euclid: euclid.kjm/1256219162
H. W. Lenstra, Jr., Rational functions invariant under a finite abelian group, Invent. Math., 25 (1974), 299–325.
Mathematical Reviews (MathSciNet): MR347788
Zentralblatt MATH: 0292.20010
Digital Object Identifier: doi:10.1007/BF01389732
T. Maeda, Noether's problem for $\mathfrak{A}_5$, J. Algebra, 125 (1989), 418–430.
Mathematical Reviews (MathSciNet): MR1018955
Zentralblatt MATH: 0697.12018
Digital Object Identifier: doi:10.1016/0021-8693(89)90174-9
J. F. Mestre, Correspondances compatibles avec une relation binaire, relèvement d'extensions de groupe de Galois $L_3(2)$ et problème de Noether pour $L_3(2)$, arXiv.org, 2005.
B. Plans, Noether's problem for $GL(2,3)$, Manuscripta Math., 124 (2007), 481–487.
Mathematical Reviews (MathSciNet): MR2357794
Zentralblatt MATH: 1146.12004
Digital Object Identifier: doi:10.1007/s00229-007-0140-0
Y. Rikuna, The existence of generic polynomial for $SL(2,3)$ over $\bm{Q}$, preprint.
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