On a Galois theory for inseparable field extensions

Heerema has developed a Galois theory for fields L of characteristic $p \ne 0$ in which the Galois subfields K are those for which $L/K$ is normal, modular and, for some nonnegative integer $e,K({L^{{p^{e + 1}}}})/K$ is separable. The related automorphism groups G are subgroups of a particular group A of automorphisms on $L[x]/{x^{{p^e} + 1}}L[x]$ where x is an indeterminate over L. For $H \subseteq G$ Galois subgroups of A, we give a necessary and sufficient condition for H to be G-invariant. An extension of a result of the classical Galois theory is also given as is a necessary and sufficient condition for every intermediate field of $L/K$ to be Galois where K is a Galois subfield of L.