  Actions of semisimple Hopf algebras H over an algebraically closed field of
characteristic zero on commutative domains were classified recently by the
authors. The answer turns out to be very simple- if the action is inner
faithful, then H has to be a group algebra. The present article contributes to
the non-semisimple case, which is much more complicated. Namely, we study
actions of finite dimensional (not necessarily semisimple) Hopf algebras on
commutative domains, particularly when H is pointed of finite Cartan type.
  The work begins by reducing to the case where H acts inner faithfully on a
field; such a Hopf algebra is referred to as Galois-theoretical. We present
examples of such Hopf algebras, which include the Taft algebras, u_q(sl_2), and
some Drinfeld twists of other small quantum groups. We also give many examples
of finite dimensional Hopf algebras which are not Galois-theoretical.
Classification results on finite dimensional pointed Galois-theoretical Hopf
algebras of finite Cartan type will be provided in the sequel, Part II, of this
study.
