Exercise 4, § 1

Show that every continuous homomorphism (necessarily injective) $f$ of the topological field $\mathbf{C}$ onto a subfield of $\mathbf{C}$ is either the identity automorphism or the automorphism $z \to \overline{z}$. [Note that we must have $f(x) = x$ for all $x \in \mathbf{Q}$, and deduce that $f(x) = x$ for all $x \in \mathbf{R}$.] Show that there exist an infinite number of discontinuous isomorphisms of $\mathbf{C}$ onto subfields of $\mathbf{C}$ other than $\mathbf{C}$ itself.