Detecting pro-p-groups that are not absolute Galois groups

Abstract

Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G F of fields F containing a primitive p th root of unity ξ p . In this paper we present several constraints on such G F , using restrictions on the cohomology of index p normal subgroups from N. Lemire, J. Mináč , and J. Swallow , Galois module structure of Galois cohomology and partial Euler-Poincaré characteristics, J. reine angew. Math. 613 (2007), 147–173. In section 1 we classify all maximal p -elementary abelian-by-order p quotients of these 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. In section 2 we derive analogues of theorems of Artin-Schreier and Becker for order p elements of certain small quotients of G F . Finally, in section 3 we construct a new family of pro- p -groups which are not absolute Galois groups over any field F .