Noether's problem for dihedral 2-groups

Abstract

Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \, g \in G)$ by $K$-automorphisms defined by $g \cdot x_h= x _{gh}$ for any $g, \, h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \, g \in G)^G$. Noethers problem asks whether $K(G)$ is rational (= purely transcendental) over $K$. We shall prove that $K(G)$ is rational over $K$ if $G$ is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.