University of Notre Dame
Browse
CousinsGD072019D.pdf (692.98 kB)

Some Model Theory of Fields and Differential Fields

Download (692.98 kB)
thesis
posted on 2019-07-03, 00:00 authored by Gregory Cousins

In this dissertation, we attempt to study connections between various properties of fields, namely boundedness, largeness, and strong form of model completeness called 'almost quantifier elimination''. The first chapter consists of background information, and the second chapter is more or less devoted to the study of fields in the ring language, possible expanded by constants. We show that large, perfect fields with almost quantifier elimination are geometric and fail to have a strong notion of unboundedness.

The third chapter is dedicated to the study, and characterization, of differential fields with a notion of largeness for differential-algebraic sets. We also make some brief comments on notions of largeness for fields equipped with an automorphism.

The fourth chapter is based on joint work with Quentin Brouette, Anand Pillay, and Françoise Point in which we prove that if T is a theory of large, bounded fields of characteristic 0 with almost quantifier elimination, and T' is the model companion of T together with the statement 'd is a derivation' then for any model (U, d) of T', differential subfield K of U such that C(K) is a model of T, and logarithmic differential equation dlog(z)=a (defined over some algebraic group G that is definable over C(K)), there is a strongly normal extension L of K for the equation with L a differential subfield of U.

History

Date Modified

2019-08-30

Defense Date

2019-06-25

CIP Code

  • 27.0101

Research Director(s)

Anand Pillay

Committee Members

Julia Knight Sergei Starchenko Peter Cholak James Freitag

Degree

  • Doctor of Philosophy

Degree Level

  • Doctoral Dissertation

Language

  • English

Alternate Identifier

1112064978

Library Record

5187110

OCLC Number

1112064978

Program Name

  • Mathematics

Usage metrics

    Dissertations

    Categories

    No categories selected

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC