Title:

Coding complete theories in Galois groups

James Ax showed that, in each characteristic, there is a natural bijection from the space of complete theories of pseudofinite fields, in first order logic, to the set of conjugacy classes of procyclic subgroups of the absolute Galois group of the prime field. I show that when the set of subgroups of a profinite group is considered to have the Vietoris (a.k.a. hyperspace, finite, exponential, neighbourhood) topology the aforementioned bijection is a homeomorphism. Thus we can think of the space of complete theories of pseudofinite fields of a given characteristic as being encoded in the absolute Galois group of the prime field. I go on to show that there is a natural way of encoding the whole space of complete theories of pseudofinite fields (i.e. without dependence on characteristic) in the absolute Galois group of the rationals. To do this I use: the theory of the algebraic padics; the relationship between the absolute Galois group of the padics and the absolute Galois group of the field with p elements; the structure of the absolute Galois group of the padics given by Iwasawa; Krasnerâ€™s lemma for henselian fields; and the Vietoris topology. At the same time, we consider the theory of algebraically closed fields with a generic automorphism (ACFA). By taking the theory of the fixed field, there is a surjective (but not injective) map from the space of complete theories of ACFA to the space of complete theories of pseudofinite fields. For the space of complete theories of ACFA, there is also a bijective Galois correspondence, in each characteristic, given by restricting the automorphism to the algebraic closure of the prime field. I show that this correspondence is a homeomorphism and that there is an analogous way of encoding the whole space in the absolute Galois group of the rationals.
