Fields of constants of infinite higher derivations

Abstract:

Let $K$ be a field of characteristic $p \ne 0$, and let $P$ be its maximal perfect subfield. Let $h$ be a subfield of $K$ containing $P$ such that $K$ is separable over $h$. We prove: Every regular subfield of $K$ containing $h$ is the field of constants of a set of higher derivations on $K$ if and only if (1) the transcendence degree of $K$ over $h$ is finite, and (2) $K$ has a separating transcendency basis over $h$. This result leads to a generalization of the Galois theory developed in [4].