I am just beginning my study in field theory, and I am doing a question about the trace of $\alpha \in K$ from $K$ to $F$ where $K,F$ are fields:
Let $K/F$ be a finite field extension and $\alpha ∈ K$. Suppose $L$ is a Galois extension of $F$ containing $K$ and $H$ is the subgroup of $\mathrm{Gal}(L/F)$ corresponding to $K$. Define the trace of $\alpha ∈ K$ from $K$ to $F$ $$ \text{Tr}_{K/F}(\alpha)= \sum_{\sigma} \sigma(\alpha)$$ where $\sigma$ ranges over all embeddings of $K$ into an algebraic closure of $F$. Then, prove that $\forall \alpha \in K, \text{Tr}_{K/F}(\alpha) \in F$.
Here, an embedding of $K$ into $\bar{F}$ means an injective homomorphism from $K$ to $\bar{F}$. And my questions are:
$1.$ I am wondering why here the problem gives me the condition concerned with $L$ and how to use them.
$2.$ My attempt to do the proof was to show that for every $\alpha$ we have $\sigma(\alpha) \in F$ but I failed. Then I was thinking about whether I can somehow show that for $\sigma$ we have $\sigma$ fixes $F$. So can someone give me some hint on this or show me other ways of proving it?
Thanks!
For your first question, note that, in order to use the Galois correspondence and related results, you need everything to be in the context of a Galois extension. That’s where the condition on $L$ comes into play - otherwise, you wouldn’t even have a bijective correspondence to define $H$, for example.
As for your second question, I’ll give you a hint: what is the fixed field of $\operatorname{Gal}(L/F)$?