Exercise 39, § 2

Let K be an ordered field and let E be an ordered algebraic extension which is maximal ordered. Show that the only K-automorphism of the field E is the identity. (First show that every K-homomorphism from E to itself is surjective, and is an isomorphism of ordered fields; if $x \in E$ has minimal polynomial $f \in K[X]$ then an automorphism of E necessarily permutes the roots of f belonging to E. Deduce that it fixes x.)