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I was reading the section on real closed fields in Jacobson's Basic Algebra II and came across a theorem (Hilbert's 17th problem) which assumed as a hypothesis that some field F is dense in its real closure.

Does anyone know any examples of ordered fields that are not dense in their real closures?

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How about rational functions $F = \mathbb R(x)$, where $x > r$ for all real $r$. The real closure contains $\sqrt{x}$, but the interval $(\sqrt{x}-1, \sqrt{x}+1)$ is disjoint from $F$.

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