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

A complete theory $T$ is strongly minimal when, in every model of $T$, every definable subset of the home sort in one variable, allowing parameters, is finite or cofinite.

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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.

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The theory ${\mathrm{ACF}}_0$ of algebraically closed fields of characteristic zero is strongly minimal in its field sort. Letand take $\kappa ={\aleph }_1$. The countable tupleshave the same type because both are algebraically independent sequences. The element $t_0$ is independent from $\bar{a}$, but there is no element of $M$ independent from $\bar{b}$, since $\bar{b}$ is a transcendence basis and $M=\mathrm{acl}(\bar{b})$. Thus the partial elementary map $\bar{a}\mapsto \bar{b}$ cannot be extended to $t_0$, so $M$ is not ${\aleph }_1$-homogeneous.

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