On Belyi pairs over arbitrary fields

The main goal of this article is to extend Grothendieck's dessins d'enfant theory to arbitrary fields. In this paper, the definitions of a Belyi pair in positive characteristic and primes of bad reduction are given. We consider the graph $K_{3,3}$. This abstract graph corresponds to three different dessins. For each dessin we find the Belyi pair and the positive characteristics for which this pair exists. The set of primes of bad reduction is also given.