  Let F be a characteristic zero differential field with an algebraically
closed field of constants, E be a no-new-constant extension of F by
antiderivatives of F and let y1, ..., yn be antiderivatives of E. The
antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the
derivatives of yi in E satisfies certain conditions. We will discuss a new
proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower
of extensions by J-I-E antiderivatives and use this generalized version of the
theorem to classify the finitely differentially generated subfields of this
tower. In the process, we will show that the J-I-E antiderivatives are
algebraically independent over the ground differential field. An example of a
J-I-E tower is extensions by iterated logarithms. We will discuss the normality
of extensions by iterated logarithms and produce an algorithm to compute its
finitely differentially generated subfields.
