Let $F$ be a field.
Recall that $\overline{F}$ is an algebraic closure of $F$ if $\overline{F}/F$ is an algebraic extension and every $f[X]\in F[X]$ can be written as a product of linear factors in $\overline{F}[X].$
It is well-known that every algebraic closure $\overline{F}$ of $F$ is algebraically closed. I am interested in its converse. More precisely.
Question: Is it true that algebraic closure $\overline{F}$ of an algebraically closed field itself? In particular, does $\overline{F} = F$ hold?