Let the field norm $N_{L/K}$ for a field extension $L/K$ be given by (for $\alpha\in L$)
$N_{L/K}(\alpha)=(\prod_{j=1}^n \sigma_j(\alpha))^{[L:K(\alpha)]}$,
where $\sigma_j(\alpha)$ are the roots of the minimal polynomial of $\alpha$ in $K$. If $L/K$ is a Galois extension, then
$N_{L/K}(\alpha)=\prod_{\sigma\in Gal(L/K)} \sigma(\alpha)$,
which is distributive over multiplication in $L$ because the automorphisms $\sigma$ are. In particular:
$N_{L/K}(\alpha\beta)=\prod_{\sigma\in Gal(L/K)} \sigma(\alpha\beta)=\prod_{\sigma\in Gal(L/K)} \sigma(\alpha)\prod_{\sigma\in Gal(L/K)}\sigma(\beta)=N_{L/K}(\alpha)N_{L/K}(\beta)$.
Question: For an arbitrary field extension $L/K$, is the norm $N_{L/k}$ distributive over multiplication?