Let $F$ be some extension of $Q_2$ (2-adic field) or $F_2((x))$ (function field). Let $E/F$ be an extension of degree $2$.

1. What is the image of the norm map $N_{E/F}$?

2. In particular - is it true that the index $[F^{\star} : N_{E/F}(E^{\star})]$ depends only on the ramification $e(E|F)$?