Abstract

A complex rational map R a a,[z] has associated its field of moduli a(3)(R), an invariant under conjugation by Mobius transformations, which is contained in every field of definition of R. In general, a(3)(R) is not a field of definition of R as it is shown by explicit examples due to Silverman. In these examples, the rational maps are definable over a degree 2 extension of the field of moduli. In this paper, we observe that such a property always holds, that is, every rational map is definable over an extension of degree at most 2 of its field of moduli. The main ingredient in the proof is Weil's descent theorem, applied to the Riemann sphere, and the fact that the group of automorphisms of degree at least 2 rational maps are well known.