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I have been going through some problems in field theory recently, and problem that I came across was the following:

"Give an example of (or show it is not possible to have) a field extension $E/F$ that is finite and a ring homomorphism $\varphi : E \longrightarrow E$ that is the identity on $F$ but is not an isomorphism."

Among the examples I tried were the map $\varphi$ that sends the imaginary part of a complex number to zero and the Frobenius Endomorphism on $\Bbb{F}_p(t)$. The first one did not work out because the map $\varphi$ was not a ring homomorphism, and the second about the Frobenius Endomorphism does not work too because $t$ is transcendental over $\Bbb{F}_p$.

So then I realised that suppose we have such a ring homomrphism from $E$ to $E$ that is the identity on $F$. Then actually $\varphi$ gives us an $F$ - linear transformation $T$ from $E$ to itself if we just look at addition now. The part I am unsure is that I want to say that the kernel of $T$ is actually trivial. I can't actually relate this to the statement that $E$ and $F$ have no proper ideals because the kernel of $T$ is neither an ideal of $E$ nor $F$. How to get around this?

The idea I am trying to use is that if the kernel is trivial, then because $E$ is finite dimensional as a vector space over $F$ by the Rank - Nullity Theorem we have the $\varphi$ is always surjective, so it is not possible to have such a $\varphi$.

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    @Benjamin: Because the two notions of kernel coincide here. This is something that needs checking if you defined kernel in terms of category theory but it's entirely obvious from the concrete definition.2012-04-06

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Right, you are correct. Namely, since $\phi$ fixes $F$ you know that $\phi$ is actually $F$-algebra map. The fact that $E$ is a field and $\phi$ a ring map tells you that $E$ is necessarily injective, but since $\phi$ is an endomorphism of a finite dimensional $F$-space which is injective it is necessarily surjective (by the Rank-Nullity theorem) and thus an isomorphism.

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    Yes, that is a common thing to be careful about--very careful.2012-04-05