In this paper we identify a class of profinite groups (totally torsion free groups) that includes all separable Galois fields containing an algebraically closed subfield, and demonstrate it can be realized as inverse limit virtually finitely generated abelian (tfvfga) groups. We show by examples the condition is quite restrictive. particular, semidirect products are rarely totally free. The res...

We study the structure of absolute differential Galois group a rational function field over an algebraically closed characteristic zero. In particular, we relate behavior embedding problems to condition that is free as proalgebraic group. Building on this, prove Matzat's freeness conjecture in case constants countably infinite transcendence degree rationals. This first known twenty year old con...

The smallest non-abelian p-groups play a fundamental role in the theory of Galois p-extensions. We illustrate this by highlighting their role in the definition of the norm residue map in Galois cohomology. We then determine how often these groups — as well as other closely related, larger p-groups — occur as Galois groups over given base fields. We show further how the appearance of some Galois...