if $K/F$ is a Galois extension, show that any intermediate field $L$ is generated by the traces of elements from $K$ over $L$.

Question

if $ K/F$ is a Galois extension, show that any intermediate field $L$ is generated by the traces of $K$ over $L$.

We know that $K$ can be generated over $F$ by a single element, say $\alpha$, I guess the trace of $\alpha$ over $L$, namely $\Sigma_{\sigma \in Gal(K/L)}(\sigma(\alpha))$, generates $L$ over $F$, but I have been unable to prove it. Any hints would be appreciated.

Answer 1

Hint: show that if $K=L(\alpha)$, then $L$ is generated over $F$ by $\text{Tr}_{K/L}(\alpha^i)$, $i\geq 0$. In order to do this, use Newton's identities to identify $M=F(\{\text{Tr}_{K/L}(\alpha^i)\})$ with the subfield $N$ of $K$ generated over $F$ by the coefficients of the minimal polynomial of $\alpha$ over $L$, then show that $N=L$.

Answer 2

Hint: take the trace of a normal basis generator.