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