Let $F$ be a field of characteristic zero. Let $D_n$ be the Weyl algebra, i.e., $D_n \subset \mathrm{End}_F(F[x_1, \dots, x_n])$ is the submodule generated by $x_i$ and $\partial_i$, $i = 1, \dots, n$.
The question is to determine whether the field of fractions $Q(F[x_1, \dots, x_n])$ of $F[x_1, \dots, x_n]$ is Noetherian as a $D_n$-module.
We know $D_n$ is a Noetherian ring. Perhaps unrelated, we proved in another part of the same problem that $F[x_1, \dots, x_n][x_1^{-1}]$ is Noetherian as a $D_n$-module, being cyclically generated by $1/x_1$.
We've been messing around a bit, but we haven't been able to find a clean solution. We think the answer is negative.
What if $F$ has characteristic $p$?
Thanks for your help.
No, it's not finitely generated, and so in particular it cannot be Noetherian. The point is that the submodule generated by any finite set of elements consists of elements where the denominator contains only finitely many possible irreducible factors in $F[x]$ (namely, the irreducible factors in the denominators of the generators), and there are infinitely many such irreducibles. (This is true over every field so this answer doesn't depend on $F$ having characteristic zero.)