Algebraically Closed Fields with a Distinct Subfield

Abstract: 


Consider the theory of pairs of ACF: algebraically closed fields with a predicate for an algebraically closed subfield. It is stable as a result of Poizat on belle pairs, and has quantifier elimination (in some language extension) by Delon.
We generalize these results to the case where the subfield is not necessarily algebraically closed, and ask what properties of the subfield transfer to the pair. Specifically, we will show quantifier elimination and model completeness results, and prove that if the subfield is NSOP1/simple/stable then so is the pair.

Joint work with Itay Kaplan and Christian d'Elbee