Exercise 4, § 2

a) Let K be a field, and P a subset of K satisfying conditions (AP_I), (AP_{II}) and (AP_{III}), and such that $K^2 \subset P$ (where $K^2$ denotes the set of squares of elements of K); show that if $x > 0$ in the ordering defined by P, then $x^{-1} > 0$; deduce that the set $K^*_+$ of elements $> 0$ in K is a subgroup of the multiplicative group of K.

b) In the field $\mathbf{Q}$, the only set $P$ which satisfies the conditions of a) is the set of rational numbers $\geqslant 0$ (in the usual ordering).

c) \* Let $P$ be a subset of $K$ satisfying (AP_I), (AP_{II}) and (AP_{III}), such that $1 \in P$ and such that $x > 0$ implies $x^{-1} > 0$ in the ordering defined by $P$. Show that $y^2 \geqslant z^2$ implies $y \geqslant z$, for positive $y$ and $z$. Deduce that, for arbitrary $y > 0$, we have

for every integer $n > 0$ (notice that $(y + y^{-1})^{2m} \geqslant \binom{2m}{m}$ for every integer $m > 0$). Deduce that, if the ordered additive group structure defined by $P$ is archimedean (VI, p. 35, Ex. 31), then $(y - z)' \geqslant 0$ for every pair of elements $y, z$ of $P$; if $K'$ is the subfield of $K$ generated by $P$ then $x^2 \geqslant 0$ for all $x \in K'$.

d) \* Let $K = R(X)$ be the field of rational functions in one indeterminate over $R$; let $P$ be the set consisting of 0 and the rational functions $u \in K$ such that $u(t)$ is defined and $> 0$ for every real number $t$. Show that $P$ satisfies the conditions of c) and generates $K$, but that there exist elements $v \in K$ such that $v^2 \notin P$.