Reduction theorems for Noether’s problem
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- by Ming-chang Kang and Bernat Plans PDF
- Proc. Amer. Math. Soc. 137 (2009), 1867-1874 Request permission
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
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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
- 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
- © Copyright 2009 American Mathematical Society
- 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
- MathSciNet review: 2480265