Introduction. Ultraproducts. Definitions and examples

The aim of this introductory chapter is to illustrate the power of the most basic principles of model theory (ultraproducts, Ƚos’s principle, Löwenheim-Skolem’s theorem and Keisler-Shelah’s ultrapower theorem) in applying them to classical questions of algebra (Hilbert’s Nullstellensatz, Hilbert’s 17^th problem, Noether-Ostrowski’s irreducibility theorem).