In this chapter we prove the main known results concerning the Diophantine classes of the rings of integers and W-integers of number fields. We start by constructing Diophantine definitions of Z over some of these rings. Next we use these definitions to put together parts of the big picture of the Diophantine classes of the rings of W-integers of number fields, discussed in Chapter 1. Most of the chapter is taken up with proving vertical results, i.e. resolving problems of the following nature. Let R1 ⊂ R2be integral domains with quotient fields F1, F2respectively, such that R2is the integral closure of R1in F2and F2/F1is a non-trivial finite field extension. Then give a Diophantine definition of R1over R2or alternatively show that R1 ≤DiophR2.

The proofs of all the vertical results presented in this book can be classified as being done by one of two vertical methods, which we name “weak” and “strong.” These methods were developed by Denef and Lipshitz in [15], [19], and [18] and consequently used by Pheidas in [68] and by the present author in [91], [99], [101], [106], [93], and [103].

Before presenting the details of the constructions for particular rings, we describe the main features of the weak and strong vertical methods.