The embedding problem over a Hilbertian PAC-field.(English)Zbl 0765.12002

A field $$P$$ is called pseudo-algebraically closed (a PAC-field) if every absolutely irreducible variety defined over $$P$$ has a $$P$$-rational point. The algebraic closure of a field $$P$$ is denoted by $$\overline{P}$$, the absolute Galois group $$G(\overline{P}/P)$$ of $$P$$ by $$G_ P$$. One says that all finite embedding problems over $$P$$ are solvable if, for every surjection $$h: E\to C$$, of finite groups and for every surjection $$\lambda: G_ P\to C$$ there exists a surjection $$\varepsilon: G_ P\to E$$ with $$h\circ \varepsilon=\lambda$$.
It is proved that every finite embedding problem over a Hilbertian PAC- field $$P$$ is solvable. For countable fields this implies that the absolute Galois group of $$P$$ is $$\omega$$-free that is $$G(\overline{P}/P)$$ is a free profinite group of countably infinite rank. It is shown that a PAC-field $$P$$ of characteristic 0 is Hilbertian if and only if all finite embedding problems over $$P$$ are solvable. A field $$P$$ is said to be regular Galois-Hilbertian if every regular (finite) Galois extension of $$P(x)$$ can be specialized to a Galois extension of $$P$$ with the same Galois group. It is proved that a PAC-field of char. 0 is RG-Hilbertian if and only if every finite group is a Galois group over $$P$$.
Reviewer: G.Pestov (Tomsk)

MSC:

 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 12F10 Separable extensions, Galois theory
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