Now, I need to discuss what happens for Kahler differentials when are fields. For simplicity, I am going to assume is characteristic zero. This is very much a cop-out. But for the sake of time, I am going to do it this way, and to please the Bourbakistas do things generally, I can always come back to talk about -bases and whatnot.

So, first suppose is a finite algebraic extension.

Proposition 1If is of characteristic zero and is a finite algebraic extension, then .

To do this, we will show that for all . Now is separable over by characteristic zero, so there is a polynomial such that . As a result

by the product rule many times, whence , proving the result.

Corollary 2If is algebraic, then .

This is now easy to prove. Given , we must prove . Now is contained in some finite-dimensional subextension , and since , a fortiori the same is true in . (more…)