Let $F$ be a field and $E$ be a finite field extension of $F$. Let's call $F$ the base field.
Do the two following proposition involving norm and trace hold?
- $\forall u \in F, N_{E/F}(u) \in F$.
- $\forall u \in F, \operatorname{Tr}_{E/F}(u) \in F$.
I have found some results like this, but it seems that always require the extension to be separable, or normal, or both. I would like to know if these proposition hold in general, just supposing that the extension is finite.