We Know that the only non zero ring Homomorphism $\phi$:$\mathbb R\to \mathbb R$,is the identity map. by this property we can show that the only nozero ring homomorphism of $\mathbb R$ is onto(Surjective). Can we have an example of a field $F$ and a nonzero homomorphism $\Psi$:$F\to F$ , that is not onto?

Note that the definition of ring homomorphism is as follows:

for all $x,y \in F, \Psi(x+y)=\Psi(x)+\Psi(y)$ and $\Psi(x.y)=\Psi(x).\Psi(y)$