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.