If $K^{alg}$ is an algebraic closure of $K$ and $L$ is a field extension, does $K^{alg}$ also an algebraic closure of $L$?

Question

My question is may be a little bit idiot, and the answers are probably obvious, but I can't find a precise answer. So let $K^{alg}$ an algebraic closure of $K$ and $L$ a field extension of $K$.

1) If $K\subset L\subset K^{alg}$, does $K^{alg}$ also an algebraic closure of $L$ ?

2) Is there so field $F$ that contain $K^{alg}$ or $K^{alg}$ is maximal ?

3) An if such a field $F$ exist, is it also algebraic or not ?

Answer 1

1) Yes,

2) Think about the algebraic closure of $\mathbb Q^{alg}$. Don't you have $\mathbb Q\subset \mathbb Q^{alg}\subset \mathbb C$ ?

3) $K^{alg}$ is maximal in the way that if $F$ is a field such that $F\supset K^{alg}$, there is an element $x\in F$ that is not algebraic.