### Abstract

We study Noether’s problem over Q for meta-cyclic groups. This paper is an extension of the previous work [2], which was concerned with the cyclic group Cn of order n. We shall give a simple description of the action of the normalizer of Cn in Sn to the function field Q(x1,…, xn), in terms of the generators of the fixed field of Cn given in [2]. Using this, we settle Noether’s problem for the dihedral group of order 2n (n ≤ 6) and the Frobenius group of order 20 with explicit construction of independent generators of the fixed fields. We shall also reconstruct some simple one-parameter families of cyclic and dihedral polynomials.

Original language | English |
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Pages (from-to) | 13-32 |

Number of pages | 20 |

Journal | Tokyo Journal of Mathematics |

Volume | 28 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 Jan 1 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Geometric generalization of gaussian period relations with application to noether’s problem for meta-cyclic groups.** / Hashimoto, Kiichiro; Hoshi, Akinari.

Research output: Contribution to journal › Article

*Tokyo Journal of Mathematics*, vol. 28, no. 1, pp. 13-32. https://doi.org/10.3836/tjm/1244208276

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TY - JOUR

T1 - Geometric generalization of gaussian period relations with application to noether’s problem for meta-cyclic groups

AU - Hashimoto, Kiichiro

AU - Hoshi, Akinari

PY - 2005/1/1

Y1 - 2005/1/1

N2 - We study Noether’s problem over Q for meta-cyclic groups. This paper is an extension of the previous work [2], which was concerned with the cyclic group Cn of order n. We shall give a simple description of the action of the normalizer of Cn in Sn to the function field Q(x1,…, xn), in terms of the generators of the fixed field of Cn given in [2]. Using this, we settle Noether’s problem for the dihedral group of order 2n (n ≤ 6) and the Frobenius group of order 20 with explicit construction of independent generators of the fixed fields. We shall also reconstruct some simple one-parameter families of cyclic and dihedral polynomials.

AB - We study Noether’s problem over Q for meta-cyclic groups. This paper is an extension of the previous work [2], which was concerned with the cyclic group Cn of order n. We shall give a simple description of the action of the normalizer of Cn in Sn to the function field Q(x1,…, xn), in terms of the generators of the fixed field of Cn given in [2]. Using this, we settle Noether’s problem for the dihedral group of order 2n (n ≤ 6) and the Frobenius group of order 20 with explicit construction of independent generators of the fixed fields. We shall also reconstruct some simple one-parameter families of cyclic and dihedral polynomials.

UR - http://www.scopus.com/inward/record.url?scp=33750537323&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750537323&partnerID=8YFLogxK

U2 - 10.3836/tjm/1244208276

DO - 10.3836/tjm/1244208276

M3 - Article

AN - SCOPUS:33750537323

VL - 28

SP - 13

EP - 32

JO - Tokyo Journal of Mathematics

JF - Tokyo Journal of Mathematics

SN - 0387-3870

IS - 1

ER -