Exercise 35, § 2

A subfield K of an ordered field E is dense in E if for every nonempty open interval $]a, b[$ in $E$ there exists $x \in K$ such that $a < x < b$. Show that E is then comparable to K (Ex. 11, a)). If E is taken to be $R(X)$, where R is a maximal ordered field and X is infinitely large with respect to R (VI, p. 43, Ex. 24), show that E is comparable to the subfield $K = R(X^2)$, but that K is not dense in E. For K to be dense in E, it is necessary and sufficient that for all $x \in E$ the partition of K consisting of $K \cap ]\leftarrow , x[$ and $K \cap (x, \to ($ be a proper cut (Ex. 34).