$K$-monomorphism $L\rightarrow L$ is $K$-automorphism if finite extension $L:K$

Question

I am trying to prove the lemma

If $L:K$ is finite then every $K$-monomorphism $L→L$ is an automorphism.

My idea is to show that every $K$-monomorphism is surjective too. Let $B=\{b_1,b_2,...,b_n\}$ a base. I have to show that elements from $B$ go to elements of $B$. I was thinking: "Is it necessary that the extension is finite?" How to prove the lemma?

Answer 1

This is a trivial since $L$ is a finite dimensional $K$-vector space, and every $K$-endomorphism $L \longrightarrow L$ is a $K$-linear map.

Now, a (well-known) result of linear algebra says that an endomophism $K^n \longrightarrow K^n$ is injective if and only if it is surjective, if and only if its determinant is $\neq 0$.

Answer 2

It is necessary that the extension be finite.

Consider $L=K(X)$, the field of rational functions over $K$.

The map $X \mapsto X^2$ induces a $K$-monomorphism $K(X) \to K(X)$ which is not surjective.