extensions of number fields - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T22:13:27Z http://mathoverflow.net/feeds/question/42703 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42703/extensions-of-number-fields extensions of number fields Oscar Villareal 2010-10-18T22:04:23Z 2010-10-18T22:41:09Z <p>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?</p> http://mathoverflow.net/questions/42703/extensions-of-number-fields/42705#42705 Answer by Felipe Voloch for extensions of number fields Felipe Voloch 2010-10-18T22:41:09Z 2010-10-18T22:41:09Z <p>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.</p>