Promoted from stack.exchange since I received no response:  
Jacobson developed a 'Galois' correspondence for purely inseperable extensions of exponent 1 (only consisting of *p*th roots) $K/k$, where he showed there was a correspondence between its subfields $k\subset{F}\subset{K}$ and finite restricted Lie sub-algebras of $k$-derivations given below:
$$F\mapsto \mathcal{D}_F(K)$$
$$\mathcal{D'}\mapsto F, \text{the field of constants of} \space \mathcal{D'}$$
Davis extends this to exponent 2 extensions, and conjectures that higher derivation Lie-algebras should correspond to higher exponents.  
Firstly, I'd like to ask for a list of books or articles that explore these ideas, especially the research done in 1968-1981, or more recently. I found the original papers in the American Mathematical Society very inaccessible, and have mostly been working off Lindsay Childs' [survey](http://pi.math.cornell.edu/~dkmiller/galmod/Childs_purely-inseparable.pdf) of this research, but would like to learn more.

The other question I have, is if this theory is applicable to the perfect closure of a field (in its algebraic closure, or more generally): is there a corresponding notion of an 'absolute lie-algebra of derivations' to that of the absolute galois group? If so please provide me with references to this too.