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

Ki Ichiro Hashimoto*, Akinari Hoshi, Yuichi Rikuna

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


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
Issue number262
Publication statusPublished - 2008 Apr
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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