Generalizations of noncommutative Noether's problem

Abstract: Noether's problem is 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 an noncommutative analogue of Noether's problem, one would need a significant skew field that shares a role similar to the field of ratioal functions. Given the importance of the Weyl fields due to Gelfand-Kirillov's Conjecture, in 2006 J. Alev and F. Dumas introduced what is nowdays called the noncommutative Noether's problem. Many papers in recent years \cite{FMO}, \cite{EFOS}, \cite{FS}, \cite{Tikaradze} have been dedicated to the subject. The aim of this article is to generalize the main result of \cite{FS} for more general versions of Noether's problem; and consider its analogue in prime characteristic.