  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 $C^n$ 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.
