Injective field homomorphism implies surjective?

Question

Suppose that $\sigma: F\to F$ is injective field homomorphism. Does it follow that $\sigma$ is surjective?

I know that is $F$ is finite field then it's true.

Is it true in general case?

Answer 1

Any field homomorphism is injective.

A simple counterexample is given by considering the field of rational functions $F(x)$ of the field $F$ over the indeterminate $x$, and the unique homomorphism $F(x)\to F(x)$ that is the identity on $F$ and maps $x$ to $x^2$.