Cogalois theory and Drinfeld modules

In this paper we generalize the results of \cite{sanchez} to rank one Drinfeld modules with class number one. We show that, in the present form, there does not exist a cogalois theory for Drinfeld modules of rank or class number larger than one. We also consider the torsion of the Carlitz module for the extension ${\mathbb{F}}_{q}(T)(\Lambda_{P^{n}}) /{\mathbb{F}}_{q}(T)(\Lambda_{P})$.