On Galois theory using pencils of higher derivations

Abstract:

Let $L \supset K$ be fields of characteristic $p \ne 0$. Assume K is the field of constants of a group of pencils of higher derivations on L, and hence L is modular over K and K is separably algebraically closed in L. Every intermediate field F which is separably algebraically closed in L and over which L is modular is the field of constants of a group of pencils of higher derivations if and only if $K({L^{{p^e}}})$ has a finite separating transcendence basis over K for some nonnegative integer e. If $p \ne 2,3$ and $K({L^{{p^e}}})$ does have a finite separating transcendence basis over K, and F is the field of constants of a group of pencils, then the group of L over F is invariant in the group of L over K if and only if $F = K({L^{{p^r}}})$ for some nonnegative integer r.