I have cross-linked with *field extension* and fixed the link to “algebraically closed”.

New entry regular field extension.

]]>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 $x$.

]]>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.

]]>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).

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