Let $k$ be a number field, and $F/k$ a finite extension. I would like to find a countable family of extensions $k\_i/k$ of degree 2 and a place $v\_i$ of $k\_i$ such that if $v$ is the place of $k$ lying below $v\_i$, then $[k\_{v\_i}:k\_v]$=2, where $k\_{v\_i}$ and $k\_v$ are the completions of $k\_i$ and $k$ at $v\_i$ and $v$. Furthermore, for some place $w$ of $F$ lying above $v$, I want the $F\_w = k\_v$. Is this possible?



extensions of number fields
There are infinitely many places $v$ of $k$ such that $F\_w=k\_v$ for some $w$ above $k$. For each such place, take $a\_v \in k, v(a\_v)=1$ and consider the extension $k(\sqrt{a\_v})/k$. It has the property you want at $v$ and you will get infinitely many such extensions as you vary $v$, so you are done.


