Model theory of algebraically closed fields

Summary

This part of model theory studies the first-order theory $\mathrm{ACF}$ of algebraically closed fields in the language of rings. It also studies the completions $\mathrm{ACF}_p$, one for each characteristic $p$. The theory $\mathrm{ACF}$ admits quantifier elimination. Each $\mathrm{ACF}_p$ is complete and decidable, and categorical in every uncountable cardinality. These facts give the Lefschetz principle and short proofs of results in algebra, such as the Ax–Grothendieck theorem. It is also the basic example of a strongly minimal theory.