Fields with continuous local elementary properties. II
The article is a continuation of Part I [Algebra Logic 33, No. 6, 351-365 (1994); translation from Algebra Logika 33, No. 6, 628-653 (1994; Zbl 0845.12005)]. The analogue of theorem 3 from that paper is proved in section 2 for the case of fields in \(RC^*_E\) (with absolutely unramified valuation rings). Such fields, in contrast to maximal fields which have ``the smallest'' possible absolute Galois group, possess a ``large'' Galois group. In section 1 some necessary group-theoretic notions are developed. Let \(G\) be a profinite group and let \(\text{Subg} G\) be the set of all closed subgroups of \(G\). \({\mathbf S} {\mathbf u} {\mathbf b} {\mathbf g} G\) endowed with the natural topology is a Boolean space. The group \(G\) acts on \({\mathbf S} {\mathbf u} {\mathbf b} {\mathbf g} G\) by conjugation. A pair \({\mathbf G} = \langle G,D_G \rangle\), where \(G\) is a profinite group and \(D_G \subseteq {\mathbf S} {\mathbf u} {\mathbf b} {\mathbf g} G\) is a closed subset such that \(\Gamma^g \in D_G\) for all \(\Gamma \in D_G\), \(g \in G\), is called a \(\Delta^*\)-group. If \({\mathbf G}_0 = \langle G_0, D_{G_0} \rangle\) and \({\mathbf G}_1 = \langle G_1, D_{G_1} \rangle\) are \(\Delta^*\)-groups, then any continuous homomorphism \(\varphi : G_0 \to G_1\) such that \(\varphi (\Gamma) \in D_{G_1}\) for every \(\Gamma \in D_{G_0}\) is called a homomorphism from \({\mathbf G}_0\) to \({\mathbf G}_1\). A homomorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) is called an epimorphism if \(\varphi : G_0 \to G_1\) is an epimorphism and \(D_{G_1} = \{\varphi (\Gamma) \mid \Gamma \in D_{G_0}\}\). An epimorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) is called flat if \(\Gamma \cap \text{Ker} \varphi = \{e\}\) for all \(\Gamma\) from \(D_{G_0}\). If \({\mathbf G}\) is a \(\Delta^*\)-group, \(X({\mathbf G})\) denotes the space \(\pi_G (D_G) \subseteq \text{Subg}^*G\), and each homomorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) of \(\Delta^*\)- groups induces a continuous map \(\varphi_*: X ({\mathbf G}_0) \to X ({\mathbf G}_1)\). A flat epimorphism \(\varphi : {\mathbf G}_0 \to {\mathbf G}_1\) of \(\Delta^*\)-groups is called narrow if \(\varphi_*\) is a homeomorphism between \(X ({\mathbf G}_0)\) and \(X ({\mathbf G}_1)\). A \(\Delta^*\)-group \({\mathbf G}\) is called universal if, for every pair of epimorphisms \(\varphi_0 : {\mathbf G} \to {\mathbf G}_0\), \(\psi : {\mathbf G}_1 \to {\mathbf G}_0\), where \({\mathbf G}_1\) is finite and \(\psi\) is narrow, there exists an epimorphism \(\varphi_1 : {\mathbf G} \to {\mathbf G}_1\), satisfying \(\psi \varphi_1 = \varphi_0\). The concept of a \(\Gamma\)-group, developed by the author in \([PC_p\)-fields with universal Galois group, Sib. Adv. Math. 1, No. 4, 1-26 (1991)], is a special case of the notion of a \(\Delta^*\)-group. In the second section the class \(uRC^*_E\) is introduced. It is the class of pairs \(\langle F,R_W \rangle\) such that \(\langle F,R_W \rangle \in RC^*_E \subset R C^*_\exists\), \(R\) is absolutely unramified for all \(R \in W\), and \({\mathbf G} (F,W)\) is a universal \(\Delta^*\)-group. Theorem. Let \(\langle F_0, R_{W_0} \rangle\), \(\langle F_1, R_{W_1} \rangle \in u RC^*_E\), \(F_1\) be a regular extension of \(F_0\), and \(\langle F_0, R_{W_0}, J(R_{W_0}) \rangle \leq \langle F_1, R_{W_1}, J(R_{W_0}) \rangle\). Then (a) if \(\langle F_0, R_{W_0}, J(R_{W_0}) \rangle \preceq \langle F_1, R_{W_1}, J(R_{W_1}) \rangle\) then \(\overline R_{W_0} \preceq \overline R_{W_1}\) and \(\Gamma_{W_0} \preceq \Gamma_{W_1}\); (b) if \(\overline R_{W_0} \preceq \overline R_{W_1}\), \(\Gamma_{W_0} \preceq \Gamma_{W_1}\), and for every \(R_0 \in W_0\), the Galois group \(G(H_{R_0} (F_0))\) is small, then \(\langle F_0, R_{W_0}, J(R_{W_0}) \rangle \preceq \langle F_1, R_{W_1}, J(R_{W_1}) \rangle\).
scientific article; zbMATH DE number 907267