Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 144 / 3 / a / i / Solution

Suppose $a\in \mathrm{acl}(A)$ in the sense of model-theoretic algebraic closure. Some formula $\phi (x,\bar{c})$ over $A$ has exactly $n<\infty$ realizations in the monster model and includes $a$. Every model $M$ containing $A$ is an elementary substructure of the monster and therefore contains $n$ distinct realizations. Since there are only $n$ in the monster, it contains all of them, including $a$. Hence