Separating $p$-bases and transcendental extension fields

Abstract:

Let $L/K$ denote an extension field of characteristic $p \ne 0$. It is known that if $L/K$ has a finite separating transcendence base, then every relative $p$-base of $L/K$ is a separating transcendence base of $L/K$. In this paper we show that when every relative $p$-base of $L/K$ is a separating transcendence base of $L/K$, then the transcendence degree of $L/K$ is finite. We also illustrate the connection between the finiteness of transcendence degree of $L/K$ and the property that $L/K(X)$ is separable algebraic for every relative $p$-base $X$ of $L/K$.