Generalizations of noncommutative Noether's problem

Abstract

Noether's problem is a classical and very important problem in algebra. It is an intrinsically interesting problem in invariant theory, but with far reaching applications in the sutdy of moduli spaces, PI-algebras, and the Inverse problem of Galois theory, among others. To obtain a noncommutative analogue of Noether's problem, one would need a significant skew field that shares a role similar to the field of rational functions. Given the importance of the Weyl fields due to Gelfand-Kirillov's Conjecture, in 2006 J. Alev and F. Dumas introduced what is nowadays called the Noncommutative Noether's problem. The aim of this article is to generalize the main result of [12] for more general versions of Noether's problem; and consider its analogue in prime characteristic.