  We present several constraints on the absolute Galois groups G_F of fields F
containing a primitive pth root of unity, using restrictions on the cohomology
of index p normal subgroups from a previous paper by three of the authors. We
first classify all maximal p-elementary abelian-by-order p quotients of such
G_F. In the case p>2, each such quotient contains a unique closed index p
elementary abelian subgroup. This seems to be the first case in which one can
completely classify nontrivial quotients of absolute Galois groups by
characteristic subgroups of normal subgroups. We then derive analogues of
theorems of Artin-Schreier and Becker for order p elements of certain small
quotients of G_F. Finally, we construct a new family of pro-p-groups which are
not absolute Galois groups over any field F.
