Universal and Existential Together in Extensions

One could argue that J. Robinson solved most of the natural first-order definability and decidability questions over number fields. Before describing this aspect of her results, we should note that in addition to the old questions of decidability and definability of Z and the rings of integers, we also have a question of uniformity of definitions across all number fields. The question of uniformity is a new question in our discussion. It naturally does not arise when we discuss Q only, and as far as existential definitions over number fields are concerned, we are very far away from being able to address such questions. However, as we will see below, the situation is different when we use the full first-order language.

In [40], J. Robinson constructed a first-order definition of Z over the ring of integers for every number field. Amazingly she used only one universal quantifier to do it. These definitions were not, however, uniform across number fields. Later on in [41], J. Robinson constructed a definition which was uniform across all number fields but was using more universal quantifiers. Rumely [45] constructed a version of these definitions uniform across global fields (number fields and function fields over a finite set of constants). Finally, In the same paper where B. Poonen constructed a two-universal-quantifier definition of Z over Q, he constructed a uniform two- universal-quantifier definition of the ring of integers across all number fields.

So far J. Koenigsmann’s one-universal-quantifler result has not been extended to any number fields, but J. Park constructed a purely universal definition of the rings of integers over all number fields in [28].

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