The rationality problem for four-dimensional linear actions

Abstract

Let $G$ be a finite subgroup of $GL(4,\mathbb{Q} )$. Let $G$ act on the rational function field $\mathbb{Q}(x_1,x_2,x_3,x_4)$ by $\mathbb{Q}$-automorphism defined by the linear action of variables.Linear Noether's problem asks whether the fixed field $\mathbb{Q} (x_1,x_2,x_3,x_4)^G$ is rational (i.e. purely transcendental) over $\mathbb{Q}$. So far some partial results have been known, but in this paper we will give the almost complete results of this problem. One of motivations of this problem is the relation to the inverse Galois problem.