A k-automorphism sigma of the rational function field k(x(1), ... , x(n)) is called purely monomial if s sends every variable xi to a monic Laurent monomial in the variables x(1), ... , x(n). Let G be a finite subgroup of purely monomial k- automorphisms of k(x(1), ... , x(n)). The rationality problem of the G-action is the problem of whether the G- fixed field k( x(1), ... , x(n)) G is k- rational, i.e., purely transcendental over k, or not. In 1994, M. Hajja and M. Kang gave a positive answer for the rationality problem of the three-dimensional purely monomial group actions except one case. We show that the remaining case is also affirmative.
展开▼
