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Abundance of Hilbertian domains. (English) Zbl 0899.12002
An integral domain \(A\) is called separably Hilbertian if every separable Hilbert set of its quotient field contains elements all of whose coordinates are in \(A\). Let \(\mathcal O\) be a countable separably Hilbertian domain with quotient field \(K\), let \(L\) be an abelian extension of \(K\) such that \(\{\text{ord}(a)\mid a\in\text{Gal}(L/K)\}\) is unbounded and let \({\mathcal O}_L\) be the integral closure of \(A\) in \(L\). Denote further by \(G(K)=G(K_s)\) the absolute Galois group of \(K\). The author shows that, for each integer \(e\geq 2\) and almost all \((\sigma_1,\dots,\sigma_e)\in G(K)^e\) (in the sense of the Haar measure), each ring between \({\mathcal O}_L\) and \(L\cdot K_s^\Sigma\) is separably Hilbertian (here \(\Sigma\) denotes the subgroup of \(G(K)\) generated by \(\sigma_1,\dots,\sigma_e\)). Denote by \(K_{\text{ab}}\) (resp. \(K_{\text{sol}}\)) the maximal abelian (resp. solvable) extension of \(K\). Then, again for \(e\geq 2\) and for almost all \((\sigma_1,\dots,\sigma_e)\in G(K)^e\), a theorem similar to the preceding one holds for all fields between \(K_{\text{ab}}\) and \(K_{s,\text{ab}}^\Sigma\) and moreover each field between \((\overline\mathbb{Q}^\Sigma\cap\overline\mathbb{Q}_{\text{tot},{\mathcal S}})_{\text{ab}}\) and \(\overline\mathbb{Q}_{\text{ab}}^\Sigma\) (where \(\overline\mathbb{Q}_{\text{tot},{\mathcal S}}\) means the maximal algebraic extension of \(\mathbb{Q}\) in which every prime of the finite set \(\mathcal S\) totally splits) has the free profinite group \(\widehat F_\omega\) of rank \(\aleph_0\) as its absolute Galois group.

MSC:
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
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