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Let $(k,<)$ be a real closed field and $L|K$ an ordered extension such that $\forall x\in L \exists y\in k\; (x.

Is $k$ dense in $L$?

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    Can't you construct a counter-example by inserting a new variable into a "gap"? e.g. if $k$ is non-archmedian then insert a new variable $T$ between the finite numbers and the positive infinite numbers, and compare $k$ to the real closure of $k(T)$.2012-07-09

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