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This problem it's from Stewart Galois Theory book. I want to solve it only using the theorems used on the book. Or at least simple tools.

Let $K$ be a field of characteristic $0$. And let $L/K$ be a finite and normal extension (thus in this case Galois extension , since is also separable). For any $a\in L$ define $T(a)= \sum_{\sigma\in G} \sigma (a)$ Where $G = Gal (L,K)$. Prove that $T(a)\in K $ and that $T$ is a surjective map $L\to K$.

Well it's clear that $T(a)\in K$ since is fixed by all the element of the galois group, and then it's just to consider the Galois theorem. But how I can prove that it's surjective?

I realized that $T$ is also a linear transformation, maybe I can use the dimension theorem is some way. But I don't know how to compare the dimension of the kernel, with the dimension of $L$ over $K$. Or maybe that it's not a good way to solve the problem.

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Hint: What is the trace of an element of $K$?

Addendum: Also it is crucial here that the characteristic of your field is $0$ as Makoto points out below. Otherwise the proof has to use slightly more sophisticated machinery.

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    Yes, your solution is great. You should post it as an answer and accept it (posting answers to your own questions is encouraged on this site). Btw, I hope you are using characteristic $0$ (you said not $0$ in your comment above) to show that the trace of $1$ is not zero, or something like that.2012-12-04