Existentially closed fields with holomorphy rings (Q1354320)

The paper studies the existence of a model companion for a theory of fields with a predicate for a subring. Precisely it is shown that the theory of fields together with an integrally closed subring, the theory of formally real fields with a real holomorphy ring and the theory of formally \(p\)-adic fields with a \(p\)-adic holomorphy have no model companions in the language of fields augmented by a unary predicate for the corresponding ring. Key facts are that any existentially closed model of each of these theories is not a field and a kind of local-global transfer principle for existentially closed models of these theories.