Model theory of algebraically closed fields and the Ax-Grothendieck Theorem

Abstract

We introduce the concept of an algebraically closed field with emphasis of the basic model-theoretic results concerning the theory of algebraically closed fields. One of these nice results about algebraically closed fields is the quantifier elimination property. We also show that the theory of algebraically closed field with a given characteristic is complete and model-complete. Finally, we introduce the beautiful Ax-Grothendieck theorem and an application to it.