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Noether's problem and $ \mathbb{Q}$-generic polynomials for the normalizer of the $ 8$-cycle in $ S_8$ and its subgroups


Authors: Ki-ichiro Hashimoto, Akinari Hoshi and Yuichi Rikuna
Journal: Math. Comp. 77 (2008), 1153-1183
MSC (2000): Primary 12F12, 14E08, 11R32.
DOI: https://doi.org/10.1090/S0025-5718-07-02094-7
Published electronically: December 3, 2007
MathSciNet review: 2373196
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Abstract: We study Noether's problem for various subgroups $ H$ of the normalizer of a group $ \mathbf{C}_8$ generated by an $ 8$-cycle in $ S_8$, the symmetric group of degree $ 8$, in three aspects according to the way they act on rational function fields, i.e., $ \mathbb{Q}(X_0,\ldots,X_7),\, \mathbb{Q}(x_1,\ldots,x_4)$, and $ \mathbb{Q}(x,y)$. We prove that it has affirmative answers for those $ H$ containing $ \mathbf{C}_8$ properly and derive a $ \mathbb{Q}$-generic polynomial with four parameters for each $ H$. On the other hand, it is known in connection to the negative answer to the same problem for $ {\mathbf C}_8/{\mathbb{Q}}$ that there does not exist a $ \mathbb{Q}$-generic polynomial for $ {\mathbf C}_8$. This leads us to the question whether and how one can describe, for a given field $ K$ of characteristic zero, the set of $ {\mathbf C}_8$-extensions $ L/K$. One of the main results of this paper gives an answer to this question.


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Additional Information

Ki-ichiro Hashimoto
Affiliation: Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3–4–1 Ohkubo, Shinjuku-ku, Tokyo, 169–8555, Japan
Email: khasimot@waseda.jp

Akinari Hoshi
Affiliation: Department of Mathematics, School of Education, Waseda University, 1–6–1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169–8050, Japan
Email: hoshi@ruri.waseda.jp

Yuichi Rikuna
Affiliation: Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3–4–1 Ohkubo, Shinjuku-ku, Tokyo, 169–8555, Japan
Email: rikuna@moegi.waseda.jp

DOI: https://doi.org/10.1090/S0025-5718-07-02094-7
Received by editor(s): October 12, 2006
Received by editor(s) in revised form: January 25, 2007
Published electronically: December 3, 2007
Article copyright: © Copyright 2007 American Mathematical Society

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