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

Published:2006-03-01
Printed: Mar 2006
• Anthony J. Bevelacqua
• Mark J. Motley
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## Abstract

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.
 Keywords: $C_i$-fields, Lang's Conjecture
 MSC Classifications: 12F - unknown classification 12F14G - unknown classification 14G

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