Exercise 2, § 9

Let $E$ be the relative algebraic closure of $\mathbf{Q}$ in $R$ ("the field of real algebraic numbers"); if $u$ is an automorphism of $E$, then $u(x) = x$ for all $x \in \mathbf{Q}$; show that if $y \in E$ is such that $y > 0$, then also $u(y) > 0$. Deduce that $u(y) = y$ for all $y \in E$. \*