HILBERTIAN FIELDS AND FREE PROFINITE GROUPS* by
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The Embedding Problem over a Hilbertian Pac-field
We show that the absolute Galois group of a countable Hilbertian P(seudo)A(lgebraically)C(losed) field of characteristic 0 is a free profinite group of countably infinite rank (Theorem A). As a consequence, G(Q̄/Q) is the extension of groups with a fairly simple structure (e.g., ∏∞ n=2 Sn) by a countably free group. In addition, we characterize those PAC fields over which every finite group is a...
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Let K be a countable separably Hilbertian field. Denote the absolute Galois group of K by G(K). For each σ ∈ (σ1, . . . , σe) ∈ G(K) let Ks[σ] be the maximal Galois extension of K which is fixed by σ1, . . . , σe. We prove that for almost all σ ∈ G(K) (in the sense of the Haar measure) the field Ks[σ] is PAC and its absolute Galois group is isomorphic to F̂ω. Mathematische Zeitschrift 224 (1997)...
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