On the Belyi degree of a number field

Abstract

In this short note we introduce the Belyi degree of a number field K, which is the smallest degree of a dessin d'enfant having K as field of moduli. After the description of some general properties (for example, the fact that there exist finitely many number fields of bounded Belyi degree), we give a lower and an upper bound for such an invariant. We finally give some explicit examples for quadratic fields.