An intermediate theory for a purely inseparable Galois theory

Abstract:

Let $K$ be a finite dimensional purely inseparable modular extension of $F$, and let $L$ be an intermediate field. This paper is concerned with an intermediate theory for the Galois theory of purely inseparable extensions using higher derivations [4]. If $L$ is a Galois intermediate field and $M$ is the field of constants of all higher derivations on $L$ over $F$, we prove that every higher derivation on $L$ over $F$ extends to $K$ if and only if $K = L{ \otimes _M}J$ for some field $J$. Similar to classical Galois theory the distinguished intermediate fields are those which are left invariant under a standard generating set for the group of all rank $t$ higher derivations on $K$ over $F$. We prove: $L$ is distinguished if and only if $L$ is $M$-homogeneous (4.9).