Specialization results in Galois theory

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The paper has three main applications. The first one is this Hilbert-Grunwald statement. If f:X→ ¹ is a degree n -cover with monodromy group Sₙ over, and finitely many suitably big primes p are given with partitions {dₚ,₁,..., dₚ,ₛₚ} of n, there exist infinitely many specializations of f at points t₀∈ that are degree n field extensions with residue degrees dₚ,₁,..., dₚ,ₛₚ at each prescribed prime p. The second one provides a description of the se-pa-ra-ble closure of a PAC field k of characteristic p=2: it is generated by all elements y such that yᵐ-y∈ k for some m≥ 2. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. A common tool is a criterion for an étale algebra ∏ₗEₗ/k over a field k to be the specialization of a k-cover f:X→ B at some point t₀∈ B(k). The question is reduced to finding k-rational points on a certain k-variety, and then studied over the various fields k of our applications.