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Definition1 - wikipedia

Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff for each $\alpha\in E$, the minimal polynomial of $\alpha$ over $F$ is separable.

Definition 2 - Lang

Let $E/F$ be an algebraic extension. Then $E/F$ is separable iff $[E:F]=[E:F]_{sep}$.

I know if the extension is finite, then these two definitions are equivalent. However, if the extension is not finite, are these definitions still equivalent? And which is the standard one?

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The statements agree when the extension is finite as you point out. However, the notion of degree is not meaningful in the context of infinite extensions.

Thus, the second characterization does not exist for infinite extensions.

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  • $\begingroup$ Why is it not meaningful as a cardinality? $\endgroup$ – Rubertos Apr 17 '15 at 17:16
  • $\begingroup$ Well, because it is infinite, and since all algebraic extensions have at most a countable dimension over the base field, it isn't a useful metric when dealing with infinite extensions. $\endgroup$ – WSL Apr 17 '15 at 18:14

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