Defining the integers in large rings of a number field using one universal quantifier

Abstract

Julia Robinson gave a first-order definition of the set of integers $$ \mathbb{Z} $$ in the rational numbers $$ \mathbb{Q} $$ by a formula (∀∃∀∃)(F = 0) where the ∀-quantifiers run over a total of 8 variables and F is polynomial. We show that for a large class of number fields, not including $$ \mathbb{Q} $$ , for every e > 0 there exist a set of primes $$ \mathcal{S} $$ of natural density exceeding 1 − e such that $$ \mathbb{Z} $$ can be defined as a subset of the “large” subring $$ \left\{ {x \in K:{\text{ord}}_p x \ge 0,\quad \forall \;\mathfrak{p} \notin \mathcal{S}} \right\} $$ of K by a formula where there is only one ∀-quantifier. In the case of $$ \mathbb{Q} $$ , we will need two quantifiers. We also show that in some cases one can define a subfield of a number field using just one universal quantifier. Bibliography: 18 titles.