Exercise 11, § 2

Let $K$ be an ordered field and $G$ a subfield of $K$. a) Show that $x \neq 0$ is infinitely large with respect to $G$ if and only if $x^{-1}$ is infinitely small with respect to $G$. We say that $K$ is comparable to $G$ if there is no element of $K$ which is infinitely large with respect to $G$ (nor, consequently, any nonzero element of $K$ which is infinitely small with respect to $G$). A necessary and sufficient condition for $K$ to be comparable to its prime subfield $\mathbf{Q}$ is that $K$ be archimedean (VI, p. 35, Ex. 31), \* and hence that $K$ be isomorphic to a subfield of $\mathbf{R}$ (VI, p. 36, Ex. 33c)). ■ b) Show that, in the ring $F(G)$ of elements of $K$ which are not infinitely large with respect to $G$, the set $I(G)$ of elements infinitely small with respect to $G$ is a maximal ideal; moreover the ordering on the quotient field $K(G) = F(G)/I(G)$ induced from that of $F(G)$ is compatible with the ring structure and is total. c) Show that a class modulo $I(G)$ can contain at most one element of $G$; deduce that the natural map from $G$ to $K(G)$ is an isomorphism from the ordered field $G$ onto a subfield $G'$ of $K(G)$, and that $K(G)$ is comparable to $G'$.