Equivalent definitions of the field norm?

Question

I had a question about the field norm $N_{E/k}$, for the finite extension $E/k$. In lecture, we defined it in my course as the determinant of the matrix for the map $\alpha \bullet: E \to E$ given by $x \mapsto \alpha x$. But apparently it is also equal to: $$ N_{E/k}(\alpha) = (\prod_{\sigma} \sigma(\alpha))^{[E:k]_i}, $$ where the product runs over all $\sigma: E \to \overline{k}$ such that $\sigma|_{k} = \mathrm{id}_k$. Can some explain from where the equivalence arises? I can't find a complete exposition of this anywhere.