Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question

1 Answer 1

up vote 4 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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