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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:
 1.2e+26 Hilbertian fields; Hilbert’s irreducibility theorem
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##### References:
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