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

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

Research output: Contribution to journalArticle

7 Citations (Scopus)

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 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
Volume2007
Issue number613
DOIs
Publication statusPublished - Dec 2007

Keywords

  • extensions
  • fields

Cite this

Detecting pro-p-groups that are not absolute Galois groups. / Benson, David John; Lemire, Nicole; Minác, Ján; Swallow, John.

In: Journal für die reine und angewandte Mathematik, Vol. 2007, No. 613, 12.2007, p. 175-191.

Research output: Contribution to journalArticle

Benson, David John ; Lemire, Nicole ; Minác, Ján ; Swallow, John. / Detecting pro-p-groups that are not absolute Galois groups. In: Journal für die reine und angewandte Mathematik. 2007 ; Vol. 2007, No. 613. pp. 175-191.
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