Rationality problem of three-dimensional monomial group actions

Let $K$ be a field of characteristic not two and $K(x,y,z)$ the rational function field over $K$ with three variables $x,y,z$. Let $G$ be a finite group of acting on $K(x,y,z)$ by monomial $K$-automorphisms. We consider the rationality problem of the fixed field $K(x,y,z)^G$ under the action of $G$, namely whether $K(x,y,z)^G$ is rational (that is, purely transcendental) over $K$ or not. We may assume that $G$ is a subgroup of $\mathrm{GL}(3,\mathbb{Z}) and the problem is determined up to conjugacy in$\mathrm{GL}(3,\mathbb{Z})$. There are 73 conjugacy classes of$G$in$\mathrm{GL}(3,\mathbb{Z})$. By results of Endo-Miyata, Voskresenski\u\i, Lenstra, Saltman, Hajja, Kang and Yamasaki, 8 conjugacy classes of 2-groups in$\mathrm{GL}(3,\mathbb{Z})$have negative answers to the problem under certain monomial actions over some base field$K$, and the necessary and sufficient condition for the rationality of$K(x,y,z)^G$over$K$is given. In this paper, we show that the fixed field$K(x,y,z)^G$under monomial action of$G$is rational over$K$except for possibly negative 8 cases of 2-groups and unknown one case of the alternating group of degree four. Moreover we give explicit transcendental bases of the fixed fields over$K$. For unknown case, we obtain an affirmative solution to the problem under some conditions. In particular, we show that if$K$is quadratically closed field then$K(x,y,z)^G$is rational over$K$. We also give an application of the result to 4-dimensional linear Noether's problem.