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Let F be a characteristic zero differential field with an algebraicallyclosed field of constants, E be a no-new-constant extension of F byantiderivatives of F and let y1, ..., yn be antiderivatives of E. Theantiderivatives y1, ..., yn of E are called J-I-E antiderivatives if thederivatives of yi in E satisfies certain conditions. We will discuss a newproof for the Kolchin-Ostrowski theorem and generalize this theorem for a towerof extensions by J-I-E antiderivatives and use this generalized version of thetheorem to classify the finitely differentially generated subfields of thistower. In the process, we will show that the J-I-E antiderivatives arealgebraically independent over the ground differential field. An example of aJ-I-E tower is extensions by iterated logarithms. We will discuss the normalityof extensions by iterated logarithms and produce an algorithm to compute itsfinitely differentially generated subfields.

Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms