Noether's problem and ℚ-generic polynomials for the normalizer of the 8-cycle in S8 and its subgroups

We study Noether’s problem for various subgroups H of the normalizer of a group C8 generated by an 8-cycle in S8, 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, . . . , x4), 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.