It is well-known that the theory of separably closed fields of some fixed positive characteristic and degree of imperfection is stable but not superstable. By a result of Cherlin and Shelah a superstable field is algebraically closed and many $\omega$-stable expansions of algebcaically fields are known ("coloured fields").


Is there an example of a non-$\omega$-stable superstable theory of (an expansion of) a field?



an example of a strictly superstable field
Hi Dima, 


The expansion of the complex field by a predicate for the set of integer powers of 2 is an example. This follows from the results of Günaydin and Van den Dries in 


"The Fields of Real and Complex Numbers with a Small Multiplicative Group"
 <http://dx.doi.org/10.1017/S0024611506015747>


Their Corollary 6.2 says that if you expand an algebraically closed field K by a predicate for a multiplicative subgroup G with the Mann property (equivalently, the Mordell-Lang property, i.e. the induced structure on the subgroup is only the group structure), then the expanded structure has the same kind of stability as the group structure on the subgroup.


In the example, the structure induced on 2^Z is just the (multiplicative) group structure (by "Mordell-Lang for G\_m"), and as a group 2^Z is isomorphic to the additive group of the integers, which is superstable, non-omega-stable. 


Also, since coloured fields were mentioned, there's a version of Poizat's green field where the coloured group is elementarily equivalent to the additive group of the integers and the structure is then superstable, non-omega-stable. This is in my (J.D. Caycedo) thesis, Section 6.5, you can find it here: <http://home.mathematik.uni-freiburg.de/caycedo/thesis>


JD


