Fields with finitely many definable subsets
Seminar Room 2, Newton Institute Gatehouse
We prove that a field with finitely many definable subsets is finite. We also conjecture a relative version of this statement: If K is a field extension of k, and the collection of sets obtained by intersecting each k-definable subset of K with K-k is finite, then k and K are either both finite or both algebraically closed. This is joint work with Kiran Kedlaya.