Discontinuous homomorphisms, selectors and automorphisms of the complex field

TL;DR

The paper demonstrates that in ZF set theory without the Axiom of Choice, the existence of a discontinuous homomorphism of real numbers implies a selector for the Vitali relation, impacting the understanding of automorphisms of the complex field.

Contribution

It proves that a nonprincipal ultrafilter on integers cannot produce a discontinuous automorphism of the complex field, confirming a conjecture of Simon Thomas.

Findings

Discontinuous homomorphisms imply Vitali selectors in ZF.

Nonprincipal ultrafilters cannot generate complex automorphisms.

Improves previous results using weaker choice assumptions.

Abstract

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.