Coding complete theories in Galois groups

• Main Author: Gray, William James Andrew
• Published: University of Edinburgh 2003
• Subjects: 530.1
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.651749

James Ax showed that, in each characteristic, there is a natural bijection from the space of complete theories of pseudo-finite 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 pseudo-finite 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 pseudo-finite 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 <i>p</i>-adics; the relationship between the absolute Galois group of the <i>p</i>-adics and the absolute Galois group of the field with <i>p</i> elements; the structure of the absolute Galois group of the <i>p</i>-adics given by Iwasawa; Krasner’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 (<i>ACFA</i>). By taking the theory of the fixed field, there is a surjective (but not injective) map from the space of complete theories of <i>ACFA</i> to the space of complete theories of pseudo-finite fields. For the space of complete theories of <i>ACFA,</i> 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.