Model Theoretic Algebra

Abstract

Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.§1. The Lefschetz principle. Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field K, and let Kn be the set of n-tuples (a1 … an) with coordinates ai in K. Kn is called affine n-space over K. Fix polynomials p1 …, pk in K[x1, …, xn] and definethat is V(p1 …, pk) is the locus of common zeroes of the pi in Kn. We call V(Pi …, Pk) the algebraic variety determined by p1, …, pk. With this terminology we may say:Algebraic geometry is the study of algebraic varieties defined over an arbitrary field K. This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.So far we have placed no restrictions on the base field K. Following Weil [4] it is convenient to start with a so-called “universal domain”; in other words take K to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.