Rationality of Three-Dimensional Quotients by Monomial Actions

Let $G$ be a finite 2-group and $K$ be a field satisfying that (i) $\text{fn}charK\mathrm{\ne }2$, and (ii) $\sqrt{a}\in K$ for any $a\in K$. If $G$ acts on the rational function field $K(x,y,z)$ by monomial $K$-automorphisms, then the fixed field $K(x,y,z{)}^G$ is rational (= purely transcendental) over $K$. Applications of this theorem will be given.