Exercise 22, § 2

Let K be a maximal ordered field and let

be a polynomial in $K[T]$ such that $a_0 \neq 0 \neq a_n$ and Show that $f$ has at most $n - 2m$ roots in K, counted with multiplicity (apply Descartes' rule). Deduce that if $g(T) = 1 + c_1 T + \ldots + c_n T^n$ has all its roots in K, and if $h(T) = 1 + b_1 T + \ldots + b_{2m} T^{2m}$ is the polynomial consisting of the first $2m + 1$ terms of the formal power series $1/g \in K[[T]]$, then h has no roots in K.