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Apply the test from Question 1. If two algebraically closed fields of the same characteristic contain a common subring $A$, they contain its fraction field and agree on all algebraic equations over it. Algebraic elements can be matched through their minimal polynomials, while transcendental elements can be matched by extending transcendence bases; an algebraic closure then gives a common extension. Thus the theory with $D(A)$ is complete. Hence the theory of algebraically closed fields of each fixed characteristic has quantifier elimination for algebraically closed fields in the ring language.

For $F\subseteq K$ and a tuple $a$ in an elementary extension, associateThis is a prime ideal. Conversely, the fraction field of $F[X]/\mathfrak{p}$ embeds into an algebraically closed extension, producing a tuple with relation ideal $\mathfrak{p}$. Quantifier elimination says this ideal determines the complete type. Thus $S_n^K(F)$ is the set of prime ideals of $F[X_1,\dots ,X_n]$. A basic formula consisting of polynomial equalities and inequalities gives a constructible subset of the prime spectrum, and these sets are clopen. This is the type space of an algebraically closed field over a subfield with its constructible topology.