Let $E/F$ be an algebraic field extension and $\bar F$ be an algebraic closure of $F$.

Define $[E:F]\_{\text{sep}}$ as the cardinality of $$\{\sigma\in \operatorname{Mono}(E,\bar F): \sigma \text{ fixes } F\}$$ (it can be proven that this is well-defined)

With this terminology, here are two definitions for separable extension:

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> **Definition 1**
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> If every minimal polynomial $m$ of $\alpha\in E$ is separable, then $E/F$ is called separable.
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> **Definition 2**
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> If $[E:F]=[E:F]\_{\text{sep}}$, then $E/F$ is called separable.
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If $E/F$ is finite, then these two definitions coincide. However, I'm not really sure why Def. 1 is stronger than Def. 2 for infinite cases. Why Def. 1 is stronger? How do I prove that? And what is the standard one?