Construction de p-extensions galoisiennes d'un corps de caractéristique différente de p. (Construction of Galois p-extensions of fields with characteristic different from p)
type: Item
instance of: scholarly article
title: Construction de p-extensions galoisiennes d'un corps de caractéristique différente de p. (Construction of Galois p-extensions of fields with characteristic different from p)
zbMATH Open document ID: 0625.12011
DOI: 10.1016/0021-8693(87)90153-0
author: Richard Massy
published in: Journal of Algebra
publication date: 1987-01-01T00:00:00Z
review text: The following type of embedding problem is explicitly solved: given an extension of fields E/K with Galois group \(G=Gal(E/K)\) an elementary abelian p-group and a group extension \[ 1 \to {\mathbb{F}}_ p \to U \to G \to 1 \tag{*} \] where \({\mathbb{F}}_ p\) denotes a cyclic group of order p, do there exist extensions of fields N/E with Gal(N/E)\(\simeq {\mathbb{F}}_ p\) and Gal(N/K)\(\simeq U\), and, if the answer is affirmative, construct these fields N. It is assumed that the characteristic of K is not p and that K contains a primitive p-th root of unity \(\zeta_ p.\) General results on embedding problems and on the Brauer group would only give the following information. A certain canonical homomorphism \(\Phi_{E/K}\) from the second cohomology group \(H^ 2(G,{\mathbb{F}}_ p)\) to the p-torsion subgroup of the Brauer group of K, \(Br(K)_ p\), has the property that the embedding problem has solutions iff the class \(\epsilon\) of (*) in \(H^ 2(G, {\mathbb{F}}_ p)\) lies in the kernel of \(\Phi_{E/K}\); moreover it is known that each element in \(Br(K)_ p\) can be written as a sum of elements of type (a,b) with \(a,b\in K^{\times 2}\), where (, ) denotes the usual symbol in \(Br(K)_ p.\) Now in this paper the author gives the following more precise results: for all \(a,b\in K^{\times}\cap E^{\times p}\) elements \((a)_ E\) and \((a,b)_ E\) in \(H^ 2(G, {\mathbb{F}}_ p)\) are defined, it is verified that \(\Phi_{E/K}((a,b)_ E)=(a,b)\) and \(\Phi_{E/K}((a)_ E)=(a,\zeta_ p)\), and it is shown that and how \(\epsilon\) can be written in a canonical way as a sum of elements \((a,b)_ E\) and \((a)_ E\) (here three cases have to be distinguished and if \(p=2\) the classification of non-degenerate quadratic forms over \({\mathbb{F}}_ 2\) is used). Furthermore the condition \(\Phi_{E/K}(\epsilon)=0\) is then rewritten in terms of a system of `norm equations' and it is shown how each solution of this system gives a solution N of the embedding problem.
reviewed by: J. Brinkhuis
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 4014850
zbMATH Keywords: p-extensions
zbMATH Keywords: embedding problem
zbMATH Keywords: Brauer group
zbMATH Keywords: symbol
zbMATH Keywords: quadratic forms over \({bbfF}_ 2\)
zbMATH Keywords: norm equations
MaRDI profile type: Publication
full work available at URL: 0021-8693(87)90153-0
OpenAlex ID: W2078554764
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description: scientific article; zbMATH DE number 4014850