For the sake of simplicity, assume the base field $k$ as having zero characteristic. I will discuss 4 different formulations of Noether's Problem.
version 1 - original Noether's problem: Let $G<S_n$ be a group that permutes the variables in $k(x_1,\ldots,x_n)$ transitively. When is $k(x_1,\ldots,x_n)^G$ a purely transcendental extension?
In case $k=\mathbb{Q}$, or any Hilbertinian field, we know that a positive solution for Noether's problem gives us a positive solution to the Inverse Problem for $G$. Moreover it implies the existence of a generic polynomial for $G$-extensions over $k$.
version 2 - linear Noether's problem: Let $G<GL_n(k)$ be a finite group acting linearly on the indeterminates in $k(x_1, \ldots, x_n)$. When is $k(x_1,\ldots,x_n)^G$ a purely transcendental extension?
This problem has a lot of applications if we assume $G$ an infinite linear algebraic group. It turns out that the question of rationality of many moduli spaces depends on the rationality of objects such as $P(V)/G$.
version 3 - multiplicative Noether's problem: let $M$ be a $G$-lattice. $G$ acts by algebra automorphisms on $k[M]$, the group algebra of $M$, which coincides with the ring of Laurient polynomials in rank $M$ indeterminates , and hence we can consider the invariants of the purely transcendental extension, $k(M)^G$. When is the invariant subfield again a purely transcendental extension?
This version is also important. The question of (stable)-rationality of the center of the division ring of $2$ $n \times n$ generic matrices, as was shown by Procesi, can be reduced to a particular case of multiplicative Noether's problem. It is one of the most important open problems in PI-algebra theory, and to the best of my knowledge, open for $n>4$.
version 4 - general Noether's problem. Let $G$ be any whatsoever finite group of automorphisms of $k(x_1,\ldots,x_n)$. When is $k(x_1,\ldots,x_n)^G$ a purely transcendental extension of $k$?
I do not know of applications of Noether's problem in this generality.
We of course have inclusions version 1 $\subset$ version 2 $\subset$ version 4; and version 1 $\subset$ version 3 $\subset$ version 4.
My question is: does versions 2, 3 or 4 also have connections to the Inverse problem in Galois theory?
