Exercise 3, § 3

a) If $f$ is an isomorphism of the field $\mathbf{R}$ onto a subfield of $\mathbf{R}$ show that $f$ must be the identity mapping of $\mathbf{R}$ onto itself. [Remark that all rational numbers are invariant under $f$ and that

so that $f$ is increasing].

b) Let $K_0$ be the field $\mathbf{R}(X)$ of rational fractions in one indeterminate over $\mathbf{R}$, ordered by taking the polynomials $> 0$ to be those whose leading coefficient is $> 0$. Let $K$ be the maximal ordered algebraic extension of $K_0$. Show that there is an order-preserving automorphism $f$ of $K$ such that $f(\xi) = \xi$ for all $\xi \in \mathbf{R}$ and $f(X) = X^2(*)$.