I'm reading Lang's "Algebra" and there's a passage in the proof of Theorem 6.3 pg.290 (namely Hilbert's Theorem 90 additive form) for which I can't find a justification, if anyone could provide an explanation, I would appreciate it.
This theorem is used for recovering Artin-Schreier Theorem for cyclic extensions in characteristic p (so, even if not explicitly stated in the theorem, the hypothesis of characteristic p can be assumed if necessary). The precise statement of the theorem is the following:
"Let $K/F$ be a cyclic field extension of degree $n$, $\sigma$ be a generator of the Galois group $\text{Gal}(K/F)$ and $\beta\in K$. Then, the trace $\text{Tr}_{K/F}(\beta)=0$ if and only if there is an element $\alpha\in K$ s.t. $\beta=\alpha-\sigma(\alpha)$."
Here, by $\text{Tr}_{K/F}(\beta)$ we mean the trace of the matrix representation of the $F$-linear map given by the multiplication by $\beta$ in $K$, seen as $F$-vector space (clearly, with respect to some fixed basis).
The unclear passage of the proof is direction $[\Rightarrow]$, in which it is claimed without any further explanation the existence of some $\theta\in K$ s.t. $\text{Tr}_{K/F}(\theta)\neq 0$. Do you see why this is the case?
Best Answer
That is because, since $K/F$ is Galois, there exists $\vartheta\in K$ such that the family $(\sigma(\vartheta))_{\sigma\in{\rm Gal}(K/F)}$ is an $F$-basis of $K$. Now, $$ {\rm Tr}_{F/K}(\vartheta)=\sum_{\sigma\in{\rm Gal}(K/F)}\sigma(\vartheta) $$ can't be $0$ because this is a non-trivial linear combination composed of vectors of a free family. Note that the fact that $K/F$ is cyclic is not used here.