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.



structure of norm one group for quadratic extension of p-adic fields
The answer will depend upon whether your quadratic extension is unramified, tamely ramified, or wildly ramified. A good place to start would be the chapter on the norm map in Fesenko-Vostokov (Ch. III in [Local fields and their extensions](https://www.maths.nottingham.ac.uk/personal/ibf/book/book.html)) or the corresponding chapter in Serre's *Corps locaux*. Both these accounts are based on a paper of Hasse from the early 1920s.


**Addendum** (2013/11/18) Dear user1832, it so happens that I was teaching exactly this topic in my course last week, and I think my notes might be useful to you. [Here they are.](https://drive.google.com/file/d/0B8EHtI8F9qdIZGVfMUV5RnhuLWM/edit?usp=sharing)


