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

Hint: take the trace of a normal basis generator.
