Let $K$ be a finite Galois extension of, say, $\mathbb{Q}$. Then is known(and called normal basis theorem) that if i view $K$ as a representation of $Gal(K/\mathbb{Q})$ over $\mathbb{Q}$ it is isomorphic to the regular representation $\mathbb{Q}[Gal(K/\mathbb{Q})]$. My question is the following:
Does anybody know a proof where one is able to directly compute the trace of each non identical element in $Gal(K/\mathbb{Q})$ , proving that is 0(of course this is equivalent to the thesis)?
One brute force attempt could be to do the calculations with primitive basis(powers of a primitive element), but the calculations seems to me quite wild.
Another little tought that i had was to write the induced map $K \otimes_{\mathbb{Q}} K$, $\phi \otimes \phi$ and using that commutes with the multiplication map, and hoping to get some infos about the square of the trace of $\phi$.
