Exercise 14, § 2

a) Let $K$ be a maximal ordered field, let $E$ be a subfield of $K$ and let $f$ be a polynomial over $E$; show that all the roots of $f$ in $K$ belong to $F(E)$ (Ex. 11, b)) (use Prop. 4 of VI, p. 24). Deduce that if $G$ is a subfield of $K$ and $E \subset K$ is an extension comparable to $G$ (Ex. 11) then the set of elements of $K$ algebraic over $E$ is a maximal ordered field comparable to $G$. b) Deduce from a) that, under the hypotheses of a), the field $K(G)$ (Ex. 11, b)) is a maximal ordered field. c) Let $f$ be a polynomial of degree $\geq 1$ over $G$. Show that $f(t)$ is infinitely large with respect to $G$ if and only if $t$ is infinitely large with respect to $G$; the element $f(t)$ is infinitely small with respect to $G$ if and only if $t$ is congruent modulo $I(G)$ to a root off in $K$ (split $K$ into intervals using the roots off and $f'$ in $K$, and apply Ex. 13; notice that if $x < t$ and if $x \in K$ is not congruent modulo $I(G)$ to $t$, then there exists $y \in K$ such that $x < y < t$ and $y - x \in G$).