Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 144 / 4 / b

In a strongly minimal theory, model-theoretic algebraic closure is a pregeometry. Given finite tuples $\bar{a},\bar{b}$ of the same type, the induced correspondence extends to an isomorphism between their algebraic closures. If $c\in \mathrm{acl}(\bar{a})$, transport it through this isomorphism. If $c\notin \mathrm{acl}(\bar{a})$, its type is the unique generic one over $\bar{a}$; choose a corresponding element outside $\mathrm{acl}(\bar{b})$. Exchange ensures that this choice has the transported type. Hence every finite partial elementary map extends, and every model is aleph-zero-homogeneous.