Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.651749
Title: Coding complete theories in Galois groups
Author: Gray, William James Andrew
Awarding Body: University of Edinburgh
Current Institution: University of Edinburgh
Date of Award: 2003
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Abstract:
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 p-adics; the relationship between the absolute Galois group of the p-adics and the absolute Galois group of the field with p elements; the structure of the absolute Galois group of the p-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 (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 pseudo-finite 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.651749  DOI: Not available
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