Exercise 38, § 2

Suppose that the field K is maximal ordered. Show that the field $\tilde{K}$ is also maximal ordered. (One can argue by contradiction, by considering an ordered algebraic extension E of $\tilde{K}$ which is maximal ordered, and supposing that $E \neq \tilde{K}$. Then there exists among the elements of $E \cap \mathbf{C}\tilde{K}$ an element w whose minimal polynomial $f \in \tilde{K}[X]$ has least degree $n > 1$. Show that there exists an interval )a, b( in E such that $a, b \in K$ and $a < w < b$, and an element $e > 0$ of E such that $f'(x) > e$ for all $x \in (a, b)$, or $f'(x) < -e$ for all $x \in (a, b)$. Show that there exists $r > 0$ in K such that $f(a) \leq -r$ and $r \leq f(b)$, or $f(a) \geq r$ and $f(b) \leq -r$. Approximate the coefficients of f by elements of K, and show that we obtain in this way a polynomial $g \in K[X]$ such that the function $x \mapsto g(x)$ is monotone in (a, b) and vanishes at some point of K in this interval, arbitrarily close to w.)