This is the last talk in our series of talks on Belyi's theorem. In this talk I will outline the proof of the fact that a compact Riemann surface S is defined over the algebraic number field if and only if the orbit of S under the Galois group Gal(C/Q) contains only finitely many isomorphism classes of Riemann surfaces. Once this is established, we will show that having a Belyi map for S leads to the finiteness of the isomorphism classes in {S^σ}_σ∈Gal(C/Q). This will conclude our study of the first three chapters of Girondo and Gonzalez-Diez's book.