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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((t)). 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.

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