This questions is mainly a question in history of mathematics, and it is a question because I am not skilled enough in german nor have the access to the relevant papers right now.
Question 1: What was the original statement of Noether's Problem? In relevant literature, I have seem both versions below:
v1: Let $G$ be a finite group, and $k(x_g)_{g \in G}$ the rational function field in the variables indexed by $G$. $G$ acts by automorphisms on this field: $g.x_h=x_{gh}$., fixing $k$. When is $k(x_g)^G$ a purely transcendental extension?
v.2 Let $G$ be a finite subgroup of $S_n$ that acts by transitive permutations on the variables in $k(x_1,\ldots,x_n)$. When is $k(x_1, \ldots, x_n)^G$ a purely transcendental extension?
Question 2: In which paper has the problem been introduced?
I've seen people quoting two different papers:
E. Noether, Rational Funktionkörper, Jahrbericht Deutsch, Math.-Verein. 22 (1913) , 316-319
And
E. Noether. Gleichungen mit vorgeschriebener Gruppe, Math. Ann. 78 (1916), 221–229
Question 3: I've read papers where people say that Noether conjectured the problem to have a positive solution to all groups $G$, and other saying that this is a myth. She did made such a Conjecture or not?
