Pencils of higher derivations of arbitrary field extensions

Abstract:

Let L be a field of characteristic $p \ne 0$. A subfield K of L is Galois if K is the field of constants of a group of pencils of higher derivations on L. Let $F \supset K$ be Galois subfields of L. Then the group of L over F is a normal subgroup of the group of L over K if and only if $F = K({L^{{p^r}}})$ for some nonnegative integer r. If $L/K$ splits as the tensor product of a purely inseparable extension and a separable extension, then the algebraic closure of K in L, $\bar K$, is also Galois in L. Given K, for every Galois extension L of K, $\bar K$ is also Galois in L if and only if $[K:{K^p}] < \infty$.