example of strongly minimal
Let be the language of rings.In other words has two constant symbols , one unary symbol , and two binary function symbols satisfying the axioms (identities) of a ring. Let be the -theory that includes the field axioms and for each the formula
which expresses that every degree polynomial which is non constant has a root. Then any model of is an algebraically closed field.
One can show that this is a complete theory and has quantifier elimination (Tarski).Thus every -definable subset of any is definable by a quantifier free formula in with one free variable
.A quantifier free formula is a Boolean combination
of atomic formulas.Each of these is of the form which defines a finite set
.Thus every definable subset of is a finite or cofinite set.Thus and are strongly minimal