# an example of a strictly superstable field

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?

-

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

-
Hi, Juan Diego! Welcome to MO and thanks for the answer. Seems like the examples of strict stability are tightly connected to the structure (Z,+); I wonder if it is a mere coincidence. – Dima Sustretov Jul 9 '12 at 13:29

As far as I know (and as far as I heard at Shelah's talk in Krakow a few days ago), the only known fields which are stable are the well-known ones:

1. The algebraically closed fields
2. Separably closed fields
3. Finite fields.

None of them satisfy your requirement.

-
Dear Goldstern, I am unsure what you mean by this statement. Surely there are known stable theories of fields that do not fit the list in the sense that their structure is not of a pure algebraically or separably closed field, e.g. the "bad field" of Baudisch-Martin-Pizarro-Hils-Wagner and all kinds of fields with "coloured points" constructed by Poizat. – Dima Sustretov Jul 8 '12 at 20:56
I may have misunderstood your question. I meant fields in the language of rings. – Goldstern Jul 8 '12 at 21:54
"A superstable non-ω-stable expansion of a field" would be a more precise term. I have updated the question. – Dima Sustretov Jul 9 '12 at 8:42