Injective endomorphism of fields

Question

Let $F$ be a field. If $\phi:F\rightarrow F$ is an injective endormorphism, can we conclude that $\phi$ is an automorphism?

Since $\phi$ is injective, we have $F\cong \phi(F)$. This implies that $\phi$ is surjective.

In many reference books, $F$ is assumed to be finite. But the above proof also holds when $F$ is infinite. Can anyone point out the mistake I made?

Answer 1

Namely, consider the map $\Phi:\Bbb Q(X)\to\Bbb Q(X)$, $\Phi(f)=f(X^2)$.

"$\phi(F)\cong F$ implies that $\phi$ is surjective" is the mistake.

Answer 2

$\Phi$ is not necessarily an automorphism. But interestingly enough, it is always injective.