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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
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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$.

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Additional Information

Ming-chang Kang
Affiliation: Department of Mathematics and Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan

Bernat Plans
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain

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

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