It is shown that any finitely generated subring of a global field has a universal first-order definition in its fraction field. This covers Koenigsmann's result for the ring of integers and its subsequent extensions to rings of integers in number fields and rings of S-integers in global function fields of odd characteristic. In this article a proof is presented which is uniform in all global fields, including the characteristic two case, where the result is entirely novel. Furthermore, the proposed method results in universal formulae requiring significantly fewer quantifiers than the formulae that can be derived through the previous approaches.

It is a problem of general interest when a domain R is first-orderdefinable within its field of fractions K via a universal first-order formula. We show that this is the case when K is a global field and R is finitely generated. Hereby we recover Koenigsmann's result that the ring of integers has a universal first-order definition in the field of rational numbers, as well as the generalisations of Koenigsmann's result by Park for number fields and by Eisenträger and Morrison for global function fields of odd characteristic.

We show that for a global field K, every ring of S-integers has a universal first-order definition in K with 10 quantifiers. We also give a proof that every finite intersection of valuation rings of K has an existential firstorder definition in K with 3 quantifiers.