## Relative regular closedness and $$\pi$$-valuations.(English. Russian original)Zbl 0791.12005

Algebra Logic 31, No. 6, 342-360 (1992); translation from Algebra Logika 31, No. 6, 592-623 (1992).
The far-reaching result of the paper is that such important classes of fields as pseudo algebraically (regularly) closed fields, pseudo $$p$$- adically closed fields, pseudo (regularly) closed fields and some others, are specific cases of a more general construction. A field $$F_ 1$$ is called a 1-extension of a field $$F_ 0$$, if $$F_ 0 \leq F_ 1$$ and for any finite set $$C \subset F_ 1$$, there exists an $$F_ 0$$-homomorphism $$\varphi$$ of the ring $$F_ 0[C]$$ into $$F_ 0$$. Let $$F$$ be a field and $$\pi$$ an element of $$F$$, $$\pi \neq 0$$, $$\pi \neq 1$$. The valuation ring $$R_ v$$ of $$F$$ is called a $$\pi$$-valuation ring, and the valuation $$v$$ a $$\pi$$-valuation, if $$v(\pi)$$ is the least positive element of the value group $$\Gamma_ v$$. A field $$F$$ is called formally $$\pi$$-adic if there exists at least one $$\pi$$-valuation of $$F$$ [Yu. L. Ershov, “Boolean families of valuation rings, Algebra Logika 31, No. 3, 276-296 (1992; see the preceding review)].
Let $$F$$ be an infinite field, $$\widetilde F$$ its algebraic closure and $$L$$ some family of intermediate fields between $$F$$ and $$\widetilde F$$. The field $$F$$ is called regularly closed with respect to a family of algebraic extensions $$L$$ if, for any regular extension $$F_ 1$$ of $$F$$, this extension is a 1-extension when $$F_ 1,F'$$ is a 1-extension of $$F'$$ for any $$F' \in L$$. If $$F$$ is regularly closed with respect to $$L$$ it is denoted by $$F\in R C (L)$$ and $$F$$ is called an $$RC(L)$$-field. Let $$W$$ be a weakly Boolean family of valuation rings for $$F$$ and let $$L_ W=\{ \sigma (H_ R(F)) \mid \sigma \in \operatorname{Aut}_ F \widetilde F,\;R \in W\}$$, where $$H_ R$$ is a henselization of $$F$$ with respect to $$R$$, $$\widetilde F$$ the algebraic closure of $$F$$. Now, let $$F$$ be a field, $$\pi \in F$$, $$\pi \neq 0,1$$. Denote by $$w_ \pi$$ the family of all $$\pi$$-valuations of $$F$$. We will write $$RC_ \pi$$ instead of $$RC(L_{w_ \pi})$$.
It is shown that all classes of fields listed in the beginning of the review are classes of $$RC(L)$$-fields under various $$L$$. The concept of a $$\pi$$-valuation and the class of $$RC_ \pi$$-fields are studied in detail. It is shown that the class of all $$RC_ \pi$$-fields is axiomatizable.
Reviewer: G.Pestov (Tomsk)

### MSC:

 12J12 Formally $$p$$-adic fields 12L12 Model theory of fields 12F05 Algebraic field extensions 12J10 Valued fields

Zbl 0791.12004
Full Text:

### References:

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