\(RC^ \ast\)-fields (Q1892821)

Most of the statements announced in the author's paper [Relatively regularly closed fields, Russ. Acad. Sci., Dokl., Math. 48, No. 2, 300- 303 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 332, No. 3, 286-288 (1993; Zbl 0824.12005)] are proved in the paper under review. A pair \(\langle F, W\rangle\) \((\langle F, R_W\rangle)\) is called an \(\text{RC}^*\)-field if \(W\) is a Boolean family of valuation rings of the field \(F\), and \(\langle F,W \rangle\) satisfies the block approximation property and a global analog of the Hensel-Rychlik property (THR). It is proved that if \(\langle F,R_W \rangle\) is an \(\text{RC}^*\)-field, then it is an RC-field (Theorem 1). If \(W\) is a weakly Boolean family of valuation rings of the field \(F\) and \(\pi: X\to W\) a continuous surjective map of Boolean spaces, then there exist a regular extension \(F_0\) of the field \(F\), a Boolean family \(W_0\) of valuation rings of \(F\), and a homeomorphism \(\varepsilon: W_0 \overset\sim {\rightarrow} X\) such that \(\langle F_0, W_0 \rangle\) is an \(\text{RC}^*\)-field, and for every \(R_0\in W_0\), \(\pi \varepsilon (R_0)= R_0\cap F\) and \(R\) is the superstructure of \(R_0 \cap F\) (Theorem 2).