A Galois theory for inseparable field extensions

Abstract:

A Galois theory is obtained for fields k of characteristic $p \ne 0$ in which the Galois subfields h are those for which k/h is normal, modular, and for some nonnegative integer r, $h({k^{{p^{r + 1}}}})/h$ is separable. The related automorphism groups G are subgroups of the group A of automorphisms $\alpha$ on $k[\bar X] = k[X]/{X^{{p^{r + 1}}}}k[X]$, X an indeterminate, such that $\alpha (\bar X) = \bar X$. A subgroup G of A is Galois if and only if G is a semidirect product of subgroups ${G_k}$ and ${G_0}$, where ${G_k}$ is a Galois group of automorphisms on k (classical separable theory) and ${G_0}$ is a Galois group of rank ${p^r}$ higher derivations on k (Jacobson-Davis purely inseparable theory). Implications of certain invariance conditions on a Galois subgroup of a Galois group are also investigated.