Exercise 17, § 2

Let K be a maximal ordered field, let $g$ be a nonzero polynomial in K[T] whose roots are all in K, and do not belong to the interval [0, n] of K. Show that for every polynomial $f(T) = a_0 + a_1 T + \ldots + a_n T^n$ of K[T], the number of roots of the polynomial

which do not belong to K, counted with multiplicity, is at most equal to the number of roots of $f$, counted with multiplicity, which do not belong to K (same method as Ex. 16).