\(\aleph_ 1\)-categorical theories were divided by Zilber into three types that can be described as follows. Let \(M\) be an uncountable model of the theory, and let \(X_ 0= \text{acl}(\emptyset)\) in \(M\). Recall that \(M\) has a strongly minimal definable set \(D\): every definable subset of \(D\) is finite or cofinite. Recall also that for a countable \(Y\subset M\), \(a\in \text{acl}(Y)\) iff the orbit of \(a\) under \(\operatorname{Aut}(M/a)\) is finite. Let \(X_ 0= \text{acl}(\emptyset)\) in \(M^{\text{eq}}\).
Type I: (“disintegrated”) \(D\) can be chosen definable over \(X_ 0\). On \(D\backslash X_ 0\) we have an equivalence relation defined by: \(\text{acl}(x)= \text{acl}(y)\). If \(\overline Y\) is any set of \(E\)- classes and \(Y= X_ 0\cup\bigcup\overline Y\) is infinite, then \(\text{acl}(Y)\) is an elementary submodel of \(M\). Conversely any elementary submodel has this form.
Type II: (“nontrivial locally modular”) \(D\) can be chosen to carry a definable Abelian group structure. \(D\) can be taken to be defined over \(X_ 0\); let \(E\) be the ring of endomorphisms of \(D/X_ 0\) induced by a definable subgroup of \(D\times D\). Then \(E\) is a division ring. \(M\) is the unique minimal prime model of \(T\) over \(D\). The natural map from \(\operatorname{Aut}(M/X_ 0)\) to \(\text{GL}(D,E)\) is surjective, and the kernel admits a finite Jordan-Hölder decomposition into profinite and Abelian groups.
Type III: All others.
It is shown in this paper that all finitely-axiomatizable \(\aleph_ 1\)- categorical theories fall into Types I and II. This is done by means of two propositions concerning the Galois theory of \(M\). If \(M\) is of type III, it is shown that arbitrarily large finite simple groups are involved in Galois groups of finite extensions within \(M\). On the other hand if \(M\) is finitely axiomatizable, only finitely many groups are so involved.
Peretyatkin has constructed examples of such theories of type I. It is not known if there are any of type II. It is conjectured that if there is one, then the division ring \(E\) mentioned in the case definition is infinite and finitely presented as a ring.