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Motivated by the classical Noether's problem, J. Alev and F. Dumas proposedthe following question, commonly referred to as the noncommutative Noether'sproblem: Let a finite group $G$ act linearly on $\mathbb{C}^n,$ inducing theaction on $\text{Frac}(A_n(\mathbb{C}))$-the skew field of fractions of the$n$-th Weyl algebra $A_n(\mathbb{C}),$ then is $\text{Frac}(A_n(\mathbb{C}))^G$isomorphic to $\text{Frac}(A_n(\mathbb{C}))?$ In this note we show that if$\text{Frac}(A_n(\mathbb{C}))^{G}\cong \text{Frac}(A_n(\mathbb{C})),$ then forany algebraically closed field $k$ of large enough characteristic, field$k(x_1,\cdots, x_n)^G$ is stably rational. This result allows us to producecounterexamples to the noncommutative Noether's problem based on well-knowncounterexamples to the Noether's problem for algebraically closed fields.

Noncommutative Noether's problem is almost equivalent to the classical Noether's problem