  This is an enhanced version of the author's 1998 Harvard Ph.D. thesis, as
published by IMRN in 2005. We propose an extension of Bogomolov's conjecture
about commutator subgroups of Galois groups to arbitrary fields. A somewhat
weaker conjecture is then shown to follow, together with the famous
Milnor-Bloch-Kato conjecture, from a certain hypothesis about homological
properties of the Milnor K-theory algebras.
