Free pseudo \(p\)-adically closed fields of finite corank
The main theorem of the paper under review extends some results of \textit{M. D. Fried} and \textit{M. Jarden} [Field arithmetic. Berlin: Springer (1986; Zbl 0625.12001)]. Let \(\bar K_1, \ldots,\bar K_{\ell}\) be \(p\)-adic closures of a field \(K\), \(\sigma_1, \ldots,\sigma_{\ell +m}\) be automorphisms in the absolute Galois group \(G(K)\) of \(K\) and \(\tilde K(\sigma_{\ell +1}, \ldots,\sigma_{\ell +m})\) be the fixed field of \(\sigma_{\ell +1}, \ldots,\sigma_{\ell +m}\) in the algebraic closure \(\tilde K\) of \(K\). The compact group \(G(K)^{\ell+m}\) is equipped with a unique normalized Haar measure (with respect to the Krull topology on \(G(K))\). The field \(K_{\sigma}=\bar K_1^{\sigma_1}\cap\cdots\cap \bar K_{\ell}^{\sigma_{\ell}}\cap \tilde K(\sigma_{\ell +1}, \ldots,\sigma_{\ell +m})\) is investigated. In order to obtain information about the \(K\)-isomorphism classes of \(p\)-adic closures of \(K\), the authors use extensively the theory of sites, developed in [\textit{D. Haran} and \textit{M. Jarden}, J. Reine Angew. Math. 383, 147--206 (1983; Zbl 0652.12010)]. It is proved that two \(\theta\)-sites on linearly disjoint extensions \(K_1,K_2\) of a field \(K\) which coincide on \(K\) extend to a \(\theta\)-site on the compositum \(K_1K_2\) (``Strong amalgamation property''). Then the class of \(P_pC\) fields with \(\ell\theta\)-sites is studied. The authors obtain a characterization theorem for such fields which resembles the characterization of \(p\)-adically closed fields in which the Hensel-Rychlik lemma holds and whose value group is a \(\mathbb Z\)-group. Finally, the main result of the paper, the intersection theorem about countable Hilbertian fields, is proved.
scientific article; zbMATH DE number 4200338
scientific article