Why is a field extension $L:K$ normal if and only if $\text{Aut}(L:K)$ acts transitively on the set of homomorphisms $L\to \overline{K}$?

Question

According to Wikipedia, given the algebraic extension $L:K$, the following are equivalent:

$a)$ The minimal polynomial over $K$ of every element in $L$ splits in $L$.

$b)$ $\text{Aut}(L:K)$ acts transitively on the set of homomorphisms $L\to \overline{K}$ where $\overline{K}$ is the algebraic closure of $K$.

I have found nothing similar to the above result (equivalence of the conditions) in Stewart's Galois Theory, and thus I'm wondering why are the conditions above equivalent?