Let $F$ be a finite extension of $Q\_2$ (2-adic field) or $F\_2((x))$ (function field). Let $E/F$ be a separable 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 $e(E|F)$?



Image of norm map for local field 
As KConrad points out, you perhaps mean to say that $F$ is a *finite* extension of $\mathbf{Q}\_2$ or of $\mathbf{F}\_2((x))$, and that the quadratic extesnions $E|F$ is *separable* (and hence galoisian) in the second case.


With this interpretation of the question, $N\_{E|F}(E^\times)$ is a closed subgroup of index $2$ in $F^\times$, and every closed subgroup of index $2$ in $F^\times$ is of this form. In particular, the ramification index $e\_{E|F}$ does *not* determine the subgroup in question.


For more on this, see the relevant chapter in Serre's *Corps locaux* (=Local fields) or the book by Fesenko and Vostokov, among many other places.


**Addendum**. In the same vein as David Speyer's example, it might also be instructive to work out the case $F=\mathbf{F}\_2((x))$,by using "Artin-Schreier theory" instead of "Kummer theory". Cyclic quadratic extensions of $F$ correspond to $\mathbf{F}\_2$-lines $D\subset F/\wp(F)$, where $\wp$ is the endomorhphism $t\mapsto t^2-t$ of the additive group of $F$. The quotient $F/\wp(F)$ carries a natural filtration, coming from the filtration on the additive group $F$, so every line $D$ has a *level* in this filtration. The ramification index of the cyclic quadratic extension $E\_D=F(\wp^{-1}(D))$ of $F$ corresponding to $D$ depends only on this level.


It would be interesting to work out the norm subgroup $N\_{E\_D|F}(E\_D^\times)\subset F^\times$ for each line $D$.


By the way, the [Second Extended Edition](http://www.maths.nott.ac.uk/personal/ibf/book/book.html) of Fesenko-Vostokov is available online.


**Addendum 2**. See [this answer](https://mathoverflow.net/questions/84274/totally-ramified-p-extension-over-f-qx/84298#84298) for a generalisation from $p=2$ to arbitrary primes $p$.


