Noether's problem and <SPAN CLASS="MATH"><IMG ></SPAN> and its subgroups期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Noether's problem and and its subgroups

Authors:	Ki-ichiro Hashimoto Akinari Hoshi Yuichi Rikuna

Institution:	Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3--4--1 Ohkubo, Shinjuku-ku, Tokyo, 169--8555, Japan ; Department of Mathematics, School of Education, Waseda University, 1--6--1 Nishi-Waseda, Shinjuku-ku, Tokyo, 169--8050, Japan ; Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3--4--1 Ohkubo, Shinjuku-ku, Tokyo, 169--8555, Japan

Abstract:	We study Noether's problem for various subgroups of the normalizer of a group $\mathbf{C}_8$ generated by an -cycle in , the symmetric group of degree , 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 containing $\mathbf{C}_8$ properly and derive a $\mathbb{Q}$ -generic polynomial with four parameters for each . 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 of characteristic zero, the set of ${\mathbf C}_8$ -extensions . One of the main results of this paper gives an answer to this question.

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