Maximal separable subfields

Abstract:

If $L/K$ is a finitely generated separable field extension of characteristic $p \ne 0$ and M is an intermediate field such that $L/M$ is inseparable, it is proved there exist subfields S of M maximal with respect to the property that $L/S$ is separable. These maximal separable subfields, denoted S-subfields for $L/M$, are characterized in two ways. (1) Let $L/S$ be a separable field extension. Then S is a S-subfield for $L/M$ if and only if $S({L^p}) \supseteq M$ and S is algebraically closed in M. (2) If $L/S$ is separable, S is a S-subfield for $L/M$ if and only if the inseparability of $L/M$ is equal to the transcendence degree of $M/S$. A S-subfield for $L/M$ is constructed using a maximal subset of a relative p-basis for $M/K$ which remains p-independent in L. It is proved that there is a unique S-subfield for $L/M$ if and only if $S/K$ is algebraic for some S.