TITLE: structure of norm one group for quadratic extension of p-adic fields
QUESTION [6 upvotes]: Let $F$ be a p-adic field (finite extensions of $\mathbb{Q}_p$ for some prime $p$), and $E/F$ be a quadratic extension. Use $\sigma$ to denote the nontrivial element in the Galois group $Gal(E/F)$. For any $x\in E$, $N(x)=x\sigma(x)$ denotes the norm of $x$.
Now consider $U(1)=\{x\in E; N(x)=1 \}$, the group of elements of norm one in $E$.We know that $U(1)$ is a subgroup of $U_{E}$, the group of units of ring of integers in $E$, thus it is a compact subgroup (correct me if I'm wrong).
I'm wondering if there are more structural results about $U(1)$? I don't have any specific meaning about 'structure', so any answer reasonable in certain sense is welcome.
Sorry for the question is not so clear, any reference is much appreciated.

REPLY [4 votes]: Aside from arithmetic point of view provided by Dalawat and Edoardo, 
let me give a group theoretical point of view about this group.
If $E/F$ is a separable quadratic field extension with the Galois group $\{id, \sigma\}$ then the map
$$\phi:E^{\times}\rightarrow U(1), \ \ x\mapsto \frac{x}{\sigma(x)}$$ is surjective (as pointed out by Edoardo). We also have $\ker\phi=F^{\times}$. In other words we have a group isomorphism  $$U(1)\simeq \frac{E^{\times}}{F^{\times}}.$$