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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

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It does not.

It is easy to prove that any algebraic extension of fields of characteristic $0$ is separable, however not every such extension is a splitting field of some polynomial - simply because all such extensions are normal, and there are some extensions of fields of characteristic $0$ which are not normal.

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