The absolute Galois group of a pseudo \(p\)-adically closed field.

*(English)*Zbl 0652.12010Fix 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 \(G(K)\cong G\).

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 \(G(K)\cong G\).

Reviewer: Şerban A. Basarab (Bucureşti)

##### MSC:

12F10 | Separable extensions, Galois theory |

12F12 | Inverse Galois theory |

11S20 | Galois theory |

20E18 | Limits, profinite groups |