The Ax-Kochen Theorem: an application of model theory to algebra

TL;DR

This paper explains the Ax-Kochen Theorem, which relates algebraic properties of p-adic numbers to model theory, providing an accessible exposition of its proof without prior logical background.

Contribution

It offers an undergraduate-level exposition of the Ax-Kochen Theorem using model theory, making the proof accessible without prior logic experience.

Findings

The Ax-Kochen Theorem connects algebraic properties of p-adic numbers with model theory.

The paper provides a clear, logic-free introduction to the theorem's proof.

It demonstrates the application of model theory to algebraic problems.

Abstract

The Ax-Kochen Theorem is a purely algebraic statement about the zeros of homogeneous polynomials over the p-adic numbers, but it was originally proved using techniques from mathematical logic. This document, the author's undergraduate honors thesis, provides an exposition of the theorem and its proof via model theory, assuming no previous experience with logic.