Projective Group Structures as Absolute Galois Structures

THE ABSOLUTE GALOIS GROUP OF A PSEUDO p-ADICALLY CLOSED FIELD

Part A. Γ-structures . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Definition of Γ-structures . . . . . . . . . . . . . . . . . . . . . . 5 2. Basic properties of Γ-structures . . . . . . . . . . . . . . . . . . . 9 3. The Γ-structure Γe,m . . . . . . . . . . . . . . . . . . . . . . . 14 4. Γ-projective groups . . . . . . . . . . . . . . . . . . . . . . . . 16 5. Projective Γ-structu...

2 4 Se p 20 02 PROJECTIVE GROUP STRUCTURES AS ABSOLUTE GALOIS STRUCTURES WITH BLOCK APPROXIMATION

Relatively projective groups as absolute Galois groups

By two well-known results, one of Ax, one of Lubotzky and van den Dries, a profinite group is projective iff it is isomorphic to the absolute Galois group of a pseudo-algebraically closed field. This paper gives an analogous characterization of relatively projective profinite groups as absolute Galois groups of regularly closed fields.

Projective extensions of fields

A field K admits proper projective extensions, i.e. Galois extensions where the Galois group is a nontrivial projective group, unless K is separably closed or K is a pythagorean formally real field without cyclic extensions of odd degree. As a consequence, it turns out that almost all absolute Galois groups decompose as proper semidirect products. We show that each local field has a unique maxi...

Projective Pairs of Profinite Groups

We generalize the notion of a projective profinite group to a projective pair of a profinite group and a closed subgroup. We establish the connection with Pseudo Algebraically Closed (PAC) extensions of PAC fields: Let M be an algebraic extension of a PAC field K. Then M/K is PAC if and only if the corresponding pair of absolute Galois groups (Gal(M),Gal(K)) is projective. Moreover any projecti...