SOME APPLICATIONS OF MODEL THEORY IN NUMBER THEORY

Abstract

In this thesis, we discuss several applications of model theory in the fields of algebraic geometry and algebraic number theory. The basic notions of first-order logic are introduced, followed by a discussion of compactness and ultraproducts. Some elementary uses of compactness are given, with emphasis on transfer principles.

Quantifier elimination is then discussed. We provide detailed proof of quantifier elimination in algebraically closed fields and real closed fields, and show some of their applications in algebraic geometry. We then demonstrate a partial quantifier elimination for Henselian valued fields, and use it to establish a transfer principle connecting p-adic fields and formal power series over finite fields. As an application we prove Ax-Kochen theorem. We also discuss some other notions in model theory.