In my textbook, the algebraic closure of a field is defined as follows:
"$\overline{F}$ is called the algebraic closure of $F$ if $\overline{F}$ is algebraic over $F$ and if every polynomial $f(x) \in F[x]$ splits completely over $\overline{F}$ (so that $\overline{F}$ can be said to contain all the elements that are algebraic over $F$)"
Can the stuff in parenthesis above be deduced from the rest of the definition?
Intuitively, this makes sense when considering the factorization of any polynomial $f(x) \in F[x]$ in $\overline{F}[x]$, but I am struggling to come up with a rigorous proof.