### Abstract

We study Noether's problem for various subgroups H of the normalizer of a group Cs generated by an 8-cycle in 5s, the symmetric group of degree 8, in three aspects according to the way they act on rational function fields, i.e., ℚ(X_{0},. . ., X_{7}), ℚ(x_{1},. . .,x _{4}), and ℚ(x, y). We prove that it has affirmative answers for those H containing C_{8} properly and derive a 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 C_{8}/ℚ that there does not exist a ℚ-generic polynomial for 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 C_{8}-extensions L/K. One of the main results of this paper gives an answer to this question.

Original language | English |
---|---|

Pages (from-to) | 1153-1183 |

Number of pages | 31 |

Journal | Mathematics of Computation |

Volume | 77 |

Issue number | 262 |

DOIs | |

Publication status | Published - 2008 Apr |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Applied Mathematics
- Computational Mathematics

### Cite this

_{8}and its subgroups.

*Mathematics of Computation*,

*77*(262), 1153-1183. https://doi.org/10.1090/S0025-5718-07-02094-7

**Noether's problem and Q-generic polynomials for the normalizer of the 8-cycle in S _{8}and its subgroups.** / Hashimoto, Kiichiro; Hoshi, Akinari; Rikuna, Yuichi.

Research output: Contribution to journal › Article

_{8}and its subgroups',

*Mathematics of Computation*, vol. 77, no. 262, pp. 1153-1183. https://doi.org/10.1090/S0025-5718-07-02094-7

_{8}and its subgroups. Mathematics of Computation. 2008 Apr;77(262):1153-1183. https://doi.org/10.1090/S0025-5718-07-02094-7

}

TY - JOUR

T1 - Noether's problem and Q-generic polynomials for the normalizer of the 8-cycle in S8and its subgroups

AU - Hashimoto, Kiichiro

AU - Hoshi, Akinari

AU - Rikuna, Yuichi

PY - 2008/4

Y1 - 2008/4

N2 - We study Noether's problem for various subgroups H of the normalizer of a group Cs generated by an 8-cycle in 5s, the symmetric group of degree 8, in three aspects according to the way they act on rational function fields, i.e., ℚ(X0,. . ., X7), ℚ(x1,. . .,x 4), and ℚ(x, y). We prove that it has affirmative answers for those H containing C8 properly and derive a 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 C8/ℚ that there does not exist a ℚ-generic polynomial for C8. This leads us to the question whether and how one can describe, for a given field K of characteristic zero, the set of C8-extensions L/K. One of the main results of this paper gives an answer to this question.

AB - We study Noether's problem for various subgroups H of the normalizer of a group Cs generated by an 8-cycle in 5s, the symmetric group of degree 8, in three aspects according to the way they act on rational function fields, i.e., ℚ(X0,. . ., X7), ℚ(x1,. . .,x 4), and ℚ(x, y). We prove that it has affirmative answers for those H containing C8 properly and derive a 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 C8/ℚ that there does not exist a ℚ-generic polynomial for C8. This leads us to the question whether and how one can describe, for a given field K of characteristic zero, the set of C8-extensions L/K. One of the main results of this paper gives an answer to this question.

UR - http://www.scopus.com/inward/record.url?scp=42349083381&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=42349083381&partnerID=8YFLogxK

U2 - 10.1090/S0025-5718-07-02094-7

DO - 10.1090/S0025-5718-07-02094-7

M3 - Article

AN - SCOPUS:42349083381

VL - 77

SP - 1153

EP - 1183

JO - Mathematics of Computation

JF - Mathematics of Computation

SN - 0025-5718

IS - 262

ER -