In a proof in my algebra book they state that it is sufficient to show that: if L/K is an algebraic extension and if $\varphi: L \rightarrow L$ is a surjective $K$-homomorphism, then $\varphi \in Aut_KL$.
I know this is true if $L/K$ is a finite algebraic extension. This because $L$ considered as vector space over $K$ has a finite dimension, and then surjectivity of $\varphi$ implies injectivity.
But how about infinite algebraic extensions? Could anyone point me into the right direction?