Exercise 6, § 2

a) Let $K$ be a lattice ordered field. Show that the following conditions are equivalent: α) $x^2 \geqslant 0$ for all $x \in K$; β) $x > 0$ implies $x^{-1} > 0$;

γ) K is strongly lattice ordered (Ex. 5); δ) K is totally ordered. (To see that β) implies δ), notice that β implies xy ≤ x² + y² for x, y ∈ K). b) Let K be the field $\mathbf{Q}(\sqrt{2})$ obtained by adjoining $\sqrt{2}$ to the field of rational numbers; as an additive group K is identified with $\mathbf{Q} \times \mathbf{Q}$ via the bijection $x + y \sqrt{2} \mapsto (x, y)$; show that if the product ordering on $\mathbf{Q} \times \mathbf{Q}$ (where Q has the usual ordering) is carried over to K via this bijection, then K is lattice ordered but not strongly lattice ordered. c) The field $K = \mathbf{Q}(X)$ of rational functions in one indeterminate X over $\mathbf{Q}$ is generated by its set of squares $K'$. Show that the set P of sums of squares of elements of K defines a non-lattice ordering on K compatible with its ring structure, and such that $u > ()$ implies $u : > 0$.