Exercise 30, § 2

a) Let K be a maximal ordered field and let m and M be two elements of K with $m < M$. Show that every polynomial $f \in K[X]$ which is positive in the interval $(m, M)$ is a sum of polynomials of the form $(\alpha X + \beta) g^2$, where $g \in K[X]$ and $\alpha X + \beta$ is positive in $(m, M)$. (Reduce to the case of first or second degree polynomials, and use the formulae

for $a < b$.) b) Show that the result of a) is no longer necessarily true when K is an arbitrary ordered field (notice that a polynomial can be > 0 in K, but not in a maximal ordered extension of K ; cf. Ex. 26, c)).