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

David John Benson, Nicole Lemire, Ján Minác, John Swallow

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


Let p be a prime. It is a fundamental problem to classify the absolute Galois groups G(F) of fields F containing a primitive pth root of unity xi(p). 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 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 GF. Finally, in section 3 we construct a new family of pro-p-groups which are not absolute Galois groups over any field F.

Original languageEnglish
Pages (from-to)175-191
Number of pages17
JournalJournal für die reine und angewandte Mathematik
Issue number613
Publication statusPublished - Dec 2007


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