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 | References | Similar Articles | Additional Information
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
MR Author ID:
714371
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
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