Why is a field extension $L:K$ normal if and only if all homomorphisms $L\to \overline{K}$ have the same image?

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 $L$.

$b)$ All homomorphisms $L\to \overline{K}$ have the same image 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?