Defining Integrality at Prime Sets of High Density in Number Fields

1. Introduction. Interest in the questions of Diophantine definability and decid-ability goes back to a question that was posed by Hilbert: Given an arbitrary polynomial equation in several variables over Z, is there a uniform algorithm to determine whether such an equation has solutions in Z? This question, otherwise known as Hilbert's tenth problem, has been answered negatively in the work of M. (see [2] and [3]). Since the time when this result was obtained, similar questions have been raised for other fields and rings. Arguably the two most interesting and difficult problems in the area are the questions of Diophantine decidability of Q and the rings of algebraic integers of arbitrary number fields. One way to resolve the question of Diophantine decidability negatively over a ring of characteristic zero is to construct a Diophantine definition of Z over such a ring. This notion is defined below.