DETECTING PRO - p - GROUPS 3

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups GF of fields F containing a primitive pth root of unity. In this paper we present several constraints on such GF , using restrictions on the cohomology of index p normal subgroups from [LMS]. In section 1 we classify all maximal p-elementary abelian-by-order p quotients of these GF . 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. In section 3 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of GF . Finally, in sections 4–6 we construct new families of pro-p-groups which are not absolute Galois groups over any field F .