Going-Down Results for $C_{i}$-Fields

http://dx.doi.org/10.4153/CMB-2006-002-5
Canad. Math. Bull. 49(2006), 11-20

  Published:2006-03-01
 Printed: Mar 2006

We search for theorems that, given a $C_i$-field $K$ and a subfield $k$ of $K$, allow us to conclude that $k$ is a $C_j$-field for some $j$. We give appropriate theorems in the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$ is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.