Quantifier elimination for algebraically closed fields

The theory of algebraically closed fields of a fixed characteristic eliminates quantifiers in the language of rings. Over a parameter field $F$, the complete type of a tuple is determined by the prime ideal of polynomial relations it satisfies.

Complete $n$-types over a subfield $F$ of an algebraically closed field correspond to prime ideals of $F[X_1,\dots ,X_n]$, or equivalently to generic points of irreducible affine varieties over $F$. The Stone topology corresponds to the constructible topology: polynomial equalities and inequalities define basic clopen sets.