Exercise 26, § 2

a) If K is an archimedean ordered field (VI, p. 40, Ex. 11, a)), and x and y are two elements of K with $x < y$, show that there exists $r \in \mathbf{Q}$ such that $x < r < y$. \* Deduce that there is only one ordered subfield of R isomorphic to K. ■ b) Consider the ordering of the field $K = \mathbf{Q}(X)$ in which X is infinitely large with respect to Q (Ex. 24). Show that K is comparable to the subfield $\mathbf{Q}(X^2)$, and give an example of two elements x, y of K such that $x < y$ and no element of $\mathbf{Q}(X^2)$ lies between x and y. c) Let K be the ordered field defined in b); show that the polynomial

over K is irreducible, that it has roots in every maximal ordered extension of K, and that $f(a) > 0$ for all $a \in K$.