Let $E$ be an algebraic extension of a field $F$.
(1) Extension degree of $E$ over $F$ is dimension of $E$ over $F$ (which is same as cardinality of basis of $E$ over $F$)
(2) Separable degree of $E$ over $F$ is the number of extension map from $E$ to algebraic closure of $F$ which extends the identity automorphism on $F$ (same as cardinality of the set of extension maps)
Since they(extension degree and separable degree) are same in finite extension case, we know they are also same as extended real sense
But in infinite dimensional case, how do we know they are really same in cardinal sense? I can’t think any clue for proving this.
(Of course, if one defines separable degree as extension degree of $S$ over $F$ where $S$ is separable closure of $F$ in $E$, we get the conclusion directly. But then, how do we know these two definitions are exactly same in cardinal sense?)
Or, Is there any counter example that they are not same in cardinal sense?