Projective pairs of profinite groups

TL;DR

This paper extends the concept of projective profinite groups to pairs involving a group and a subgroup, linking it to PAC extensions of fields, and constructs new examples of such extensions.

Contribution

It introduces the notion of projective pairs of profinite groups and connects them to PAC field extensions, providing new constructions and characterizations.

Findings

A projective pair corresponds to a PAC extension of a PAC field.

Any projective pair can be realized as Galois groups of a PAC extension.

Constructed new examples of PAC extensions, including unbounded abelian extensions of Q.

Abstract

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 projective pair can be realized as absolute Galois groups of a PAC extension of a PAC field. Using this characterization we construct new examples of PAC extensions of relatively small fields, e.g., unbounded abelian extensions of the rational numbers.