Show if $α ∈ L −$ {$0$} then $N(α) ∈ F − ${$0$}

Question

Let $L$ be Galois over $F$ and $G = Gal(L|F)$. Let $α ∈ L − {0}$. Define the norm by $N(α) = \prod _{σ∈G} σ(α)$. I wish to show that if $α ∈ L −$ {$0$} then $N(α) ∈ F − ${$0$} and if $α, β ∈ L$, then $N(αβ) = N(α)N(β)$.

For the second part, we have $N(αβ) = \prod _{σ∈G} σ(αβ) = \prod _{σ∈G} σ(α)σ(β)=\prod _{σ∈G} σ(α)\prod _{σ∈G} σ(β)=N(α)N(β)$ Right? Hmm but for the first part, I am not sure ~

Answer 1

If $\alpha \neq 0$, then $\forall \sigma Gal(L/F), \sigma(\alpha) \neq 0$. So $N(\alpha) \neq 0$. And it belongs to $F$ because it is fixed by $Gal(L/F)$ : $\tau(N(\alpha))=\prod_{\sigma \in Gal(L/F)} \tau \sigma (\alpha)=\prod_{\sigma \in Gal(L/F)} \sigma (\alpha)=N(\alpha)$.