Does a separable field imply splitting field?
I am thinking yes. For $F < E$ If every $\alpha \in E$ is separable over $F$ then it must be that $E$ is a $S.F.$ over $F$?
Also we have the condition that the index of $E$ over $F, \ \{E:F\} $ divides the degree of $E$ over $F$, $[E:F]$ and when $E$ is separable, these two are equal.
Thank you