Any $K$-homomorphism is an automorphism.

Question

Let $F/K$ be an algebraic extension.Then any $K$-homomorphism from $F$ to $F$ is an automorphism.

Clearly the $K$-homomorphism is one to one as it is defined from a field to a field.So the only requirement is to show that it onto.

Any insight. Thanks in advance.

Answer 1

Let $a$ be in F with minimal polynomial f, and S is the set containing all conjugations of a in F (there are only finite many of them as they are all roots of f). Now the morphism preserves S and is injective hence must be surjective on S as it's finite. So $a$ lies in the image hence it's onto.