A simple remark on the field of moduli of rational maps (Q2922440)

A complex rational map has the form \(R(z) = P(z)/Q(z)\), where \(P(z), Q(z) \in \mathbb{C}[z]\) are coprime polynomials with complex coefficients. Each complex rational map \(R\) has associated its moduli field \(\mathcal{M}_R\) which is contained in every field of definition of \(R\). On the contrary, in general \(\mathcal{M}_R\) is not a field of definition of \(R\), since \textit{J. H. Silverman} found counterexamples in [Compos. Math. 98, No. 3, 269--304 (1995; Zbl 0849.11090)].NEWLINENEWLINEIn this paper the author proves that every rational map can be defined either over its field of moduli, or over an extension of degree \(2\) of its field of moduli.