4. Galois Actions

This chapter first collects basic material about Galois theory for finite and infinite field extensions, with examples chosen from number fields and function fields. The latter examples provide a link between Galois groups and covering groups for regular coverings. Another important example is the absolute Galois group \(\mathbb{G}\), the automorphism group of the field of all algebraic numbers: as the projective limit of the (finite) Galois groups of the Galois extensions of the rationals, this is a profinite group, with a natural topology, the Krull topology, making it a topological group. Belyĭ’s Theorem implies that \(\mathbb{G}\) has a natural action on dessins, through its action on the algebraic numbers defining them. As observed by Grothendieck, this action is faithful, so it gives a useful insight into the Galois theory of algebraic number fields. In the second section, moduli fields of algebraic curves are defined, and we discuss their relation to fields of definition. Weil’s cocycle condition is explained. We sketch two proofs of the other direction of Belyĭ’s theorem, that a curve can be defined over an algebraic number field if it admits a Belyĭ function. We list some Galois invariants and non-invariants of dessins, which are useful in determining orbits of \(\mathbb{G}\), and we give a proof due to Lenstra and Schneps that \(\mathbb{G}\) acts faithfully on the set of dessins formed from trees in the plane.