# Monomorphisms of a finite field extension

I have to show that if $L:K$ is a finite field extension and we have a $K$-monomorphism then this is an automorphism. I'm a bit confused by this if we have a k monomorphism:

$f_K:L\rightarrow G$ for some field $G$ and we have that $f(k)=k$ for all $k\in K$ I can't see why this leads to it being an automorphism?

Thanks for any help

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The notion of "automorphism" does only make sense for endomorphisms (see azimut's answer), otherwise there are trivial counterexamples (see DonAntonio's answer). –  Martin Brandenburg Mar 6 '13 at 17:56

## 2 Answers

This is false unless $\,L/K\,$ is a normal extension. For example, the monomorphism

$$\phi:\Bbb Q(\sqrt[4]2)\to\overline{\Bbb Q}\;\;,\;\;\phi(\sqrt[4]2):=\sqrt[4]2i$$

is not an automorphism, and $\,\phi(q)=q\,\,,\,\,\,\forall\,q\in\Bbb Q\,$

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Since $L$ is a $K$-vector space of finite dimension, a $K$-linear map $L\to L$ is injective if and only if it is surjective.

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