Differential Operators on Finite Purely Inseparable Extensions

We study the the differential operators of a finite modular field extension. Using the Jacobson-Bourbaki Theorem, we establish criteria for when a subalgebra of the differential operators on an extension corresponds to an intermediate modular extension. Furthermore, we can determine when an extension is modular using a sequence of modules of differentials. Finally, this thesis will clarify and expand on Gerstenhaber's theory of higher derivations and their correspondences with modular extensions, and we determine criteria for when a subspace of the symbol algebra corresponds to an intermediate extension in a simple example.