Reduction theorems for Noether's problem

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 and $h\cdot x(g)=x(hg)$. Denote by $K(G)=K(x(g):g\in G)^G$ the fixed field. Noether's problem asks whether $K(G)$ is rational (= purely transcendental) over $K$. We will give several reduction theorems for solving Noether's problem. For example, let $\widetilde{G}=G\times H$ be a direct product of finite groups. Theorem. Assume that $K(H)$ is rational over $K$. Then $K(\widetilde{G})$ is rational over $K(G)$. In particular, if $K(G)$ is rational (resp. retract rational) over $K$, so is $K(\widetilde{G})$ over $K$.