Bases consisting of norms and traces

$$\newcommand{\gal}{\text{Gal}}$$ I am trying to give a detailed proof of the following facts:

If $$L$$ is an intermediate field of a Galois extension $$K/F$$, then it is generated by traces or norms of $$K$$ over $$L$$. In other words $$L=F(\{T_{K/L}(a)|a\in K\})=F(\{N_{K/L}(a)|a\in K\}).$$

For the first fact:

$$\textit{Proof}.$$ Using the normal basis theorem, $$\exists \alpha\in K$$ such that $$\{\sigma(\alpha)|\sigma\in \gal(K/L)\}$$ is a basis for $$K$$ over $$L$$. Now, let $$\beta=T_{K/L}(\alpha)$$ and $$\tau\in \gal(K/F)$$.

I claim, $$\tau(\beta)=\beta$$ if and only if $$\tau\in \gal(K/L)$$. When $$\tau\in \gal(K/L)$$, clearly $$\tau(T_{K/L}(\alpha))=\sum_{\sigma\in \gal(K/L)}\tau\sigma(\alpha)=\sum_{\sigma\in\gal(K/L)}\sigma(\alpha)=T_{K/L}(\alpha).$$

Here's the first part I'm unsure of:

I know that $$\gal(K/L)$$ is precisely the subgroup of $$\gal(K/F)$$ fixing the subfield $$L$$ and so if we suppose $$\tau\notin \gal(K/L)$$ then $$\exists a\in L$$ with $$\tau(a)\notin L$$. Say $$\sigma(\alpha)=a$$. Then

$$\{\tau\sigma(\alpha)|\sigma\in \gal(K/L)\}\neq\{\sigma(\alpha)|\sigma\in \gal(K/L)\}$$ and hence $$\tau(\beta)=\sum_{\sigma\in \gal(K/L)}\tau\sigma(\alpha)\neq\sum_{\sigma\in\gal(K/L)}\sigma(\alpha)=\beta.$$

Then, by the fundamental theorem of Galois Theory, $$F(\beta)=L$$. $$\hspace{5cm}\square$$?

For the second fact:

$$\textit{Proof.}$$ First we use the primitive element theorem to write $$K=F(\alpha)$$. Let $$f(x)=\min(\alpha,L)=\prod_{\sigma\in \gal(K/L)}(x-\sigma(\alpha))=x_n+a_{n-1}x^{n-1}+...+a_1x+a_0.$$ Now $$L=F(\{a_i\}_{i=1}^{n-1})$$ and we want to show that $$F(\{a_i\}_{i=1}^{n-1})=F(\{T_{K/L}(a)|a\in K\})$$. Choose any $$b\in F$$ and define $$a:=b-\alpha$$ $$N_{K/F}(a)=\prod_{\sigma\in \gal(K/L)}\sigma(a)=\prod_{\sigma\in\gal(K/F)}\sigma(b)-\sigma(\alpha)=\prod_{\sigma\in \gal(K/F)}(b-\sigma(\alpha))=f(b)$$

Here's the second part I am unsure of:

Now I want to show that $$F(\{f(b)|b\in F\})=F(\{a_i\}_{i=1}^{n-1})$$ but I am unsure how.

To summarize, here are the precise questions:

-How can I show that $$F(\{f(b)|b\in F\})=F(\{a_i\}_{i=1}^{n-1})$$?

-Is my argument to show that $$\tau(\beta)=\beta\Leftrightarrow \tau\in \gal(K/L)$$ valid?

-How do we know $$L=F(\{a_i\})$$?

• In characteristic $0$ it is clear $Tr_{K/L}$ is surjective $K \to L$. For the norm you can say that $a \in L$ is the $N-1$-th derivative of $(a-x)^N$ which is $\sum_{j=0}^{N-1} {N-1 \choose j} (-1)^j (a+j)^N = \sum_{j=0}^{N-1} { N-1 \choose j} (-1)^j N_{K/L}(a+j)$. – reuns Aug 19 at 1:16