Proposition 5

— Let K be a maximal ordered field, and let f be a polynomial in K[X] which changes sign between two elements a and b of K (with $a < b$). Then f has a root x in K such that $a < x < b$.

At least one of the irreducible factors off, say $h$, changes sign between $a$ and $b$. Then the field $K[X]/(h)$ admits the structure of an ordered extension of $K$ (VI, p. 23, Prop. 3), and $h$ has degree 1 (Def. 4). Since $h(a) h(b) < 0$, the unique root $x$ of $h$ is such that $a < x < b$, since a polynomial function of degree 1 is monotonic.