Noether’s problem and $\mathbb {Q}$-generic polynomials for the normalizer of the $8$-cycle in $S_8$ and its subgroups

We study Noether’s problem for various subgroups $H$ of the normalizer of a group $\mathbf {C}_8$ generated by an $8$-cycle in $S_8$, the symmetric group of degree $8$, 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 $H$ containing $\mathbf {C}_8$ properly and derive a $\mathbb {Q}$-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 ${\mathbf C}_8/{\mathbb {Q}}$ 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 $K$ of characteristic zero, the set of ${\mathbf C}_8$-extensions $L/K$. One of the main results of this paper gives an answer to this question.