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

Published electronically:
January 6, 2009

MathSciNet review:
2480265

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be any field, and be a finite group. Let act on the rational function field by -automorphisms and . Denote by the fixed field. Noether's problem asks whether is rational (= purely transcendental) over . We will give several reduction theorems for solving Noether's problem. For example, let be a direct product of finite groups. Theorem. Assume that is rational over . Then is rational over . In particular, if is rational (resp. retract rational) over , so is over .

**1.**Hamza Ahmad, Mowaffaq Hajja, and Ming-chang Kang,*Rationality of some projective linear actions*, J. Algebra**228**(2000), no. 2, 643–658. MR**1764585**, 10.1006/jabr.2000.8292**2.**Mowaffaq Hajja,*Rational invariants of meta-abelian groups of linear automorphisms*, J. Algebra**80**(1983), no. 2, 295–305. MR**691805**, 10.1016/0021-8693(83)90002-9**3.**Mowaffaq Hajja and Ming Chang Kang,*Some actions of symmetric groups*, J. Algebra**177**(1995), no. 2, 511–535. MR**1355213**, 10.1006/jabr.1995.1310**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.**Hideo Kuniyoshi,*On a problem of Chevalley*, Nagoya Math. J.**8**(1955), 65–67. MR**0069160****6.**Bernat Plans,*Noether’s problem for 𝐺𝐿(2,3)*, Manuscripta Math.**124**(2007), no. 4, 481–487. MR**2357794**, 10.1007/s00229-007-0140-0**7.**Joseph J. Rotman,*An introduction to the theory of groups*, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR**1307623****8.**David J. Saltman,*Generic Galois extensions and problems in field theory*, Adv. in Math.**43**(1982), no. 3, 250–283. MR**648801**, 10.1016/0001-8708(82)90036-6**9.**David J. Saltman,*Retract rational fields and cyclic Galois extensions*, Israel J. Math.**47**(1984), no. 2-3, 165–215. MR**738167**, 10.1007/BF02760515**10.**Bhama Srinivasan and Judith D. Sally (eds.),*Emmy Noether in Bryn Mawr*, Springer-Verlag, New York-Berlin, 1983. MR**713788**

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