Exercise 36, § 2

Let K be an ordered field and let $\tilde{K}$ be the set of major sets D such that the cut (S, D) is proper ; show that $\tilde{K}$ is an ordered field, taking addition to be that of the monoid $\mathfrak{M}(K)$. and the product of two elements $D_1$ and D, of $\tilde{K}$ contained in K, to be the set $\langle D_1 D_2 \rangle$ (VI, p. 35, Ex. 30), where $D_1 D_2$ denotes the set of $x_1 x_2$ for $x_i \in D_i$ and $x_2 \in D_2$. Show that for any increasing homomorphism f from K onto a dense subfield F of an ordered field E, there exists an increasing homomorphism g from E into $\tilde{K}$ such that $g \circ f$ is the natural injection of K into $\tilde{K}$.