  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.
