Instead of defining it as a subfield of algebraic closure, what would be the natural way to define it?
Here's what I figured out:
Let $F$ be a field and $E$ be an extension field of $F$.
Then, $E$ is called a separable closure of $F$ iff $E/F$ is separable and for any separable extension $L$ of $E$, $L=E$.
Is this a correct definition? So that a separable closure means a maximal separable extension?
Secondly, I'm now curious why the standard definition of algebraic closure is defined so. Here is the standard definition of an algebraic closure.
Let $F$ be a field and $E$ be a field extension of $F$.
Then, $E$ is called an algebraic closure of $F$ iff $E/F$ is algebraic and $E$ is algebraically closed.
Rather than the above definition, I think the below one is more natural:
Let $F$ be a field and $E$ be a field extension of $F$.
Then, $E$ is an algebraic closure of $F$ iff $E/F$ is algebraic and for any algebraic extension $L$ of $E$, $L=E$.
Under this definition, one can see "closure" merely means a maximal object in a partially ordered category, hence I think this is more natural.. What's a reason for the standard definition is defined as so?