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$\newcommand{\gal}{\text{Gal}}$ 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 $L=F(\{a_i\})$?

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  • $\begingroup$ 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 $\sum_{j=0}^{N-1} {N-1 \choose j} (-1)^j (a+j)^N = \sum_{j=0}^{N-1} { N-1 \choose j} (-1)^j N_{K/L}(a+j)$. $\endgroup$ – reuns Aug 19 at 1:16

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