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Chevalley’s theorem on constructible sets and elimination of quantifiers. The entries are related ! The interest came partly from teaching some classical algebraic geometry these days. The related entry is also forking, though yet it is not said why; non-forking may be viewed as related to a notion of generic point, generic type (in the sense of model theory).
Of course they’re related! Taking the image is the same as applying an existential quantifier.
Compare the magnificent theorem of Tarski-Seidenberg, which is more or less the theorem that the theory of the real numbers as ordered field admits elimination of quantifiers and is decidable.
I would like to see (I am convinced there are many) a model categorical generalization of the following: look at the notion of the generic point in the sense of Weil of an irreducible affine variety over a field $k$. The generic point in the sense of Weil of the variety is point of affine space over the universal domain $\Omega$ (algebraically closed field which is of infinite transcendence over k) with the property that the variety is its locus in the sense of having the same annihilator in $k[X_1,\ldots,X_n]$. Now a variety has a generic point $x$ in the sense of Weil iff $k(x)$ is a regular field extension of the field $x$.
New entry regular field extension.
I have cross-linked with field extension and fixed the link to “algebraically closed”.
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