0
$\begingroup$

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?

$\endgroup$

1 Answer 1

4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.