Given an \(\mathcal{L}\)-structure \(\mathfrak{M}=\langle M,\sigma\rangle\) where \(\mathcal{L}\) is a countable language. A definable set \(A\subseteq M^n\) is called strongly minimal if for every elementarily equivalent extension \(\mathfrak{M}'\) of \(\mathfrak{M}\), every definable subset \(B \subseteq A(\mathfrak{M}')\) is either finite or cofinite. A complete theory \(T\) is called strongly minimal if for every \(\mathcal{L}\)-structure \(\mathfrak{M} \models T\) and every definable subset \(A \subseteq M^1\) is finite of cofinite.
Theorem (Morley)
Let \(\kappa\) be an uncountable cardinal and \(T\) be a complete theory. \(T\) has only one model of cardinality \(\kappa\) up to isormorphism if and only if \(T\) has only one model of cardinality \(\lambda\) up to isormorphism for every uncountable \(\lambda\).
Let \(T\) be a strongly minimal theory. \(T\) is called disintegrated if \({\rm acl}(A) = \cup_{a \in A} {\rm acl}(a)\) for all \(A \subseteq \mathfrak{M}\) where \(\mathfrak{M}\implies T\) is some saturated model.
Theorem (Zilber)
If \(X\) is strongly minimal, then one of the following holds
\(X\) is trivial in the sense that \({\rm acl}\) is disintegrated.
\(X\) is essentially a vector space possibly after adding some constant symbols to the language \(\mathcal{L}\).
There is a type-definable pseudoplane interpretable in \(X\).