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)


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


Zbl 0791.12004
Full Text: DOI


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