  We show, in Zermelo-Fraenkel set theory without the Axiom of Choice, that the
existence of a discontinuous homomorphism of the additive group of real numbers
induces a selector for the Vitali equivalence relation $\mathbb{R}/\mathbb{Q}$.
This shows that a nonprincipal ultrafilter on the integers is not sufficient to
construct a discontinuous automorphism of the complex field, confirming a
conjecture of Simon Thomas. This is an improved version of our paper in the
Proceedings of the American Mathematical Society, which used a weak version of
the Axiom of Choice for the same result.
