Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 4 / a / Solution

In a strongly minimal theory, model-theoretic algebraic closure is a pregeometry. If $C$ is algebraically closed, every element outside $C$ realizes the generic type in a strongly minimal theory: every one-variable definable set is finite or cofinite, and a point outside $C$ belongs to none of the finite ones.

Enumerate a finite tuple $a_1,\dots ,a_n$ of distinct elements of the independent set $A$. Independence saysIts image tuple under the bijection has the same property. Inductively, $a_i$ and $f(a_i)$ realize the same generic type over the algebraic closures of the preceding tuples, so every finite restriction of $f$ is elementary. First-order formulas involve only finitely many parameters, hence the entire bijection is an elementary map. This is the independent set in a strongly minimal theory principle.