Twisted covers and specializations

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The central topic is this question: is a given k-étale algebra ∏ₗEₗ/k the specialization of a given k-cover f:X→ B at some point t₀∈ B(k)? Our main tool is a twisting lemma that reduces the problem to finding k-rational points on a certain k-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over PAC or ample fields.