let $L=F(S),$ where $S=\{x_1,x_2, \ldots \}$ is an infinite set of algebraically independent elements over $F.$ let $K$ be the algebraic closure of $L$ and define the $F$-homomorphism $\varphi: L \longrightarrow K$ by $\varphi(x_n)=x_{n+1}, \ n \geq 1.$ since $K$ is algebraically closed, a Zorn's lemma argument shows that $\varphi$ can be extended to an $F$-homomorphism $\psi: K \longrightarrow K.$ now $\psi$ is not surjective because, for example, if $\psi(u)=x_1$ for some $u \in K,$ then, since $u$ is algebraic over $L,$ $x_1$ would be algebraic over $\psi(L)=\varphi(L) = F(S \setminus \{x_1\}).$ but that is impossible because then $x_1$ and $S \setminus \{x_1\}$ would be algebraically dependent.