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

Kiichiro Hashimoto, Akinari Hoshi, Yuichi Rikuna

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    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(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.

    Original languageEnglish
    Pages (from-to)1153-1183
    Number of pages31
    JournalMathematics of Computation
    Volume77
    Issue number262
    DOIs
    Publication statusPublished - 2008 Apr

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    ASJC Scopus subject areas

    • Algebra and Number Theory
    • Applied Mathematics
    • Computational Mathematics

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