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

Fried, Michael D.; Völklein, Helmut

Ann. Math. (2) 135, No. 3, 469-481 (1992).

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

Keywords:

pseudo-algebraically closed field; regular Galois-Hilbertian field; finite embedding problem; Hilbertian PAC-field; absolute Galois group; free profinite group