Given a differential ring $R$, I will discuss how to classify differential extensions $\phi :R^{\prime}\rightarrow R$ where $R^{\prime}$ is a differential ring and $\phi$ is a differential surjective map with a kernel $I$ such that $I^2=0$. I will first discuss the differential structure of such extensions and determine a condition that guarantees a differential splitting for $\phi $ after a differential extension $R\rightarrow S$. Then I will outline a new Grothendieck topology for differential rings such that locally these square zero extensions are differentially split. As a consequence $H^{0},$ $H^{1},$ and $H^{2}$ may be identified with automorphisms of such an extension, a principal homogeneous space for all such differential extensions, and an obstruction space to the existence of a such differential extension with kernel $I$. In order to carry out this program I will need to sketch Andre's approach to differential algebra.

This is joint work undertaken with Carlos Arreche.