Introduction

Abstract

In the middle of the 1960s, several model theoretic arguments and methods of construction captured the attention of the mathematical world. J. Ax and S. Kochen succeeded, in joint work, in achieving a decisive contribution to the “Artin conjecture” on the solvability of homogeneous diophantine equations over p-adic number fields. This and other results led to an infiltration of certain model theoretic concepts and methods into algebra. Because of their strangeness, however, very few algebraists could become comfortable with them. This is not so astonishing once one traces the historical development of model theoretic concepts and methods from their origin up to their present-day applications: their applicability to algebra is not the result of a goal-directed development (more precisely, a development directed toward the goal of these applications), but is, rather, a by-product of an investigation directed toward a quite different goal, namely, the goal of getting to grips with the foundations of mathematics. The need to secure these foundations had become urgent after various contradictions in mathematics were discovered at the end of the nineteenth and the beginning of the twentieth centuries.