Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Reduction theorems for Noether's problem


Authors: Ming-chang Kang and Bernat Plans
Journal: Proc. Amer. Math. Soc. 137 (2009), 1867-1874
MSC (2000): Primary 12F12, 12F20, 13A50, 11R32, 14E08
DOI: https://doi.org/10.1090/S0002-9939-09-09608-7
Published electronically: January 6, 2009
MathSciNet review: 2480265
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ K$ be any field, and $ G$ be a finite group. Let $ G$ act on the rational function field $ K(x(g):g\in G)$ by $ K$-automorphisms and $ h\cdot x(g)=x(hg)$. Denote by $ K(G)=K(x(g):g\in G)^G$ the fixed field. Noether's problem asks whether $ K(G)$ is rational (= purely transcendental) over $ K$. We will give several reduction theorems for solving Noether's problem. For example, let $ \widetilde{G}=G\times H$ be a direct product of finite groups. Theorem. Assume that $ K(H)$ is rational over $ K$. Then $ K(\widetilde{G})$ is rational over $ K(G)$. In particular, if $ K(G)$ is rational (resp. retract rational) over $ K$, so is $ K(\widetilde{G})$ over $ K$.


References [Enhancements On Off] (What's this?)

  • 1. H. Ahmad, M. Hajja and M. Kang, Rationality of some projective linear actions, J. Algebra 228 (2000), 643-658. MR 1764585 (2001e:12003)
  • 2. M. Hajja, Rational invariants of meta-abelian groups of linear automorphisms, J. Algebra 80 (1983), 295-305. MR 691805 (84g:20010)
  • 3. M. Hajja and M. Kang, Some actions of symmetric groups, J. Algebra 177 (1995), 511-535. MR 1355213 (96i:20013)
  • 4. M. Kang, Actions of dihedral groups, in ``A festschrift in honor of Prof. Man-Keung Siu, 2005'', Hong Kong University Press, to appear.
  • 5. H. Kuniyoshi, On a problem of Chevalley, Nagoya Math. J. 8 (1955), 65-67. MR 0069160 (16:993d)
  • 6. B. Plans, Noether's problem for $ GL(2,3)$, Manuscripta Math. 124 (2007), 481-487. MR 2357794 (2008i:12009)
  • 7. J. J. Rotman, An introduction to the theory of groups, Graduate Texts in Math. 148, Springer-Verlag, New York, 1995 (fourth edition). MR 1307623 (95m:20001)
  • 8. D. J. Saltman, Generic Galois extensions and problems in field theory, Advances in Math. 43 (1982), 250-283. MR 648801 (84a:13007)
  • 9. D. J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), 165-215. MR 738167 (85j:13008)
  • 10. R. G. Swan, Noether's problem in Galois theory, in ``Emmy Noether in Bryn Mawr'', edited by B. Srinivasan and J. Sally, Springer-Verlag, Berlin, 1983. MR 713788 (84g:01001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 12F12, 12F20, 13A50, 11R32, 14E08

Retrieve articles in all journals with MSC (2000): 12F12, 12F20, 13A50, 11R32, 14E08


Additional Information

Ming-chang Kang
Affiliation: Department of Mathematics and Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan
Email: kang@math.ntu.edu.tw

Bernat Plans
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain
Email: bernat.plans@upc.edu

DOI: https://doi.org/10.1090/S0002-9939-09-09608-7
Keywords: Noether's problem, rationality problem, retract rational.
Received by editor(s): August 29, 2007
Received by editor(s) in revised form: March 7, 2008
Published electronically: January 6, 2009
Additional Notes: The second-named author was partially supported by MTM2006-04895 (Ministerio de Educación y Ciencia) and by 2005SGR00557 (Generalitat de Catalunya).
Communicated by: Martin Lorenz
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society