Galois theory for fields $K/k$ finitely generated

Abstract:

Let K be a field of characteristic $p \ne 0$. A subgroup G of the group ${H^t}(K)$ of rank t higher derivations $(t \leq \infty )$ is Galois if G is the group of all d in ${H^t}(K)$ having a given subfield h in its field of constants where K is finitely generated over h. We prove: G is Galois if and only if it is the closed group (in the higher derivation topology) generated over K by a finite, abelian, independent normal iterative set F of higher derivations or equivalently, if and only if it is a closed group generated by a normal subset possessing a dual basis. If $t < \infty$ the higher derivation topology is discrete. M. Sweedler has shown that, in this case, h is a Galois subfield if and only if $K/h$ is finite modular and purely inseparable. Also, the characterization of Galois groups for $t < \infty$ is closely related to the Galois theory announced by Gerstenhaber and and Zaromp. In the case $t = \infty$, a subfield h is Galois if and only if $K/h$ is regular. Among the applications made are the following: (1) ${ \cap _n}h({K^{{p^n}}})$ is the separable algebraic closure of h in K, and (2) if $K/h$ is algebraically closed, $K/h$ is regular if and only if $K/h({K^{{p^n}}})$ is modular for $n > 0$.