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The absolute Galois group of a pseudo $$p$$-adically closed field. (English) Zbl 0652.12010
Fix a prime number $$p$$. A field $$K$$ of characteristic zero is called pseudo $$p$$-adically closed (PpC) if every absolutely irreducible variety $$V$$ defined over $$K$$ has a $$K$$-rational point, provided $$V$$ has a $$\tilde K$$-rational point for each $$p$$-adic closure $$\tilde K$$ of $$K$$. The main goal of this interesting paper is to describe the absolute Galois group $$G(K)$$ of such a field $$K$$.
Let $$\Gamma =G({\mathbb Q}_ p)$$ be the absolute Galois group of the field $${\mathbb Q}_ p$$ of $$p$$-adic numbers. This profinite group admits a description by generators and relations [cf. U. Jannsen and K. Wingberg, Invent. Math. 70, 71–98 (1982; Zbl 0534.12010) and K. Wingberg, ibid., 99–113 (1982; Zbl 0534.12011)]. For a profinite group $$G$$, a pair ($$\alpha: B\to A,\, \phi: G\to A$$), where $$\alpha$$ is an epi of finite groups and $$\phi$$ is a (continuous) morphism, is called a $$\Gamma$$-embedding problem for $$G$$ if for each closed subgroup $$H$$ of $$G$$ which is isomorphic to $$\Gamma$$ there exists a morphism $$\gamma_ H: H\to B$$ such that $$\alpha \circ \gamma =\text{Res}_ H\phi$$. The $$\Gamma$$-embedding problem ($$\alpha, \phi$$) is solvable if there exists a morphism $$\gamma_ H: G\to B$$ such that $$\alpha \circ \gamma =\phi$$. $$G$$ is called $$\Gamma$$-projective if every finite $$\Gamma$$-embedding problem for $$G$$ is solvable and if the collection of all closed subgroups of $$G$$ which are isomorphic to $$\Gamma$$ is topologically closed.
The authors prove the following basic result:
Theorem: If $$K$$ is PpC then $$G(K)$$ is $$\Gamma$$-projective. Conversely, if the profinite group $$G$$ is $$\Gamma$$-projective then there exists a PpG field $$K$$ such that $$G(K)\cong G$$.

##### MSC:
 12F10 Separable extensions, Galois theory 12F12 Inverse Galois theory 11S20 Galois theory 20E18 Limits, profinite groups
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