Let $E/F$ be a field extension with $E$ algebraically closed. Show that every $F$-embedding $E \to E$ is an automorphism if and only if ${\rm tr. deg}(E/F ) < \infty$.

**Sufficiency**: We have a $F$-automorphism of $F(B)$ ($B$ is a transcendence basis of $E/F$), say $\varphi$, since $E$ is an algebraic closure of $F(B)$, $\varphi$ extends to a $F$-automorphism of $E$.

Is My proof correct? How to prove the necessity? Thanks for your help.