Small maximal pro-\(p\) Galois groups
type: Item
instance of: scholarly article
title: Small maximal pro-\(p\) Galois groups
zbMATH Open document ID: 0902.12003
author: Ido Efrat
published in: Manuscripta Mathematica
publication date: 1998-12-07T00:00:00Z
full work available at URL: https://eudml.org/doc/156373
review text: Let \(p\) be an odd prime, and let \(K\) be a field of characteristic not \(p\) and containing a primitive \(p\)th root of unity. Let \(G_K(p)\) be the maximal pro-\(p\) Galois group of \(K\), i.e. the Galois group over \(K\) of the field \(K(p)\) which is the composite inside a fixed algebraic closure of all finite Galois \(p\)-extensions of \(K\). This paper provides a classification of all such groups which are (topologically) generated by at most \(4\) elements, with the exception of the pro-\(p\) Demuškin groups on \(4\) generators. A similar classification is already known for \(p = 2\) and up to \(5\) generators, via the classification of abstract Witt rings with at most \(2^5\) square classes. The computations in this paper are based on classifying \(p\)-quaternionic pairings of small order; these pairings were introduced by \textit{Y. Hwang} and \textit{B. Jacob} [Can. J. Math. 47, 527-543 (1995; Zbl 0857.12002)].
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 13K05
zbMATH DE Number: 1151666
zbMATH Keywords: Galois \(p\)-extensions
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: \(p\)-quaternionic pairings
MaRDI profile type: Publication
OpenAlex ID: W2066728047
DOI: 10.1007/S002290050026
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description: scientific article; zbMATH DE number 1151666