The equivalence of the Kohn finite ideal type and the D’Angelo finite type with the subellipticity of the \(\bar{\partial }\) -Neumann problem is extended to pseudoconvex domains in \({\mathbb {C}}^n\) whose defining function is in a Denjoy-Carleman quasianalytic class closed under differentiation. The proof involves algebraic geometry over a ring of germs of Denjoy-Carleman quasianalytic functions that is not known to be Noetherian and that is intermediate between the ring of germs of real analytic functions and the ring of germs of smooth functions. It is also shown that this type of ring of germs of Denjoy-Carleman functions satisfies the \(\sqrt{acc}\) property, one of the strongest properties a non-Noetherian ring could possess.
