Galois cohomology of biquadratic extensions
type: Item
instance of: scholarly article
title: Galois cohomology of biquadratic extensions
zbMATH Open document ID: 0781.12005
DOI: 10.1007/BF02565813
published in: Commentarii Mathematici Helvetici
publication date: 1994-02-21T00:00:00Z
full work available at URL: https://eudml.org/doc/140297
review text: Let \(F\) be a field of characteristic different from 2, and let \(\Gamma\) be the Galois group of the separable algebraic closure \(\overline{F}\) over \(F\). Let \(\Lambda\) be the field of two elements, considered as a \(\Gamma\)-module by the trivial action of \(\Gamma\); and if \(K\) is a subfield of \(\overline{F}\) which is a normal, finite dimensional extension of \(F\), let \(H^ n K\) denote the cohomology group \(H^ n(\Gamma',\Lambda)\) for the corresponding subgroup \(\Gamma'\) of \(\Gamma\). Note that \(H^ n K\) is isomorphic to \(H^ n(\Gamma, \Lambda[\Gamma/\Gamma'])\) for the group algebra of the Galois group \(\Gamma/\Gamma'\) of \(K\) over \(F\). Now let \(M=F[\sqrt{a_ 1}, \sqrt{a_ 2}, \sqrt{a_ 3}]\) be a biquadratic extension of \(F\) for elements \(a_ 1\), \(a_ 2\), \(a_ 3\) of \(F\) such that \(a_ 1 a_ 2 a_ 3=1\), and assume that \(\sqrt{a_ 1} \sqrt{a_ 2} \sqrt{a_ 3}=1\). Let \(G\) be the Galois group of \(M\) over \(F\), and for \(i=1,2,3\), let \(L_ i=F[\sqrt{a_ i}]\) and let \(G_ i\) be the Galois group of \(L_ i\) over \(F\). The authors construct a long exact sequence of \(\Gamma\)-modules: \[ 0 \longrightarrow \Lambda \overset \text{res} \longrightarrow \Lambda[G] \longrightarrow \bigoplus ^ 3_{i=1} \Lambda[G_ i] \longrightarrow \Lambda[G] \overset \text{cor} \longrightarrow \Lambda, \] where res maps 1 to the sum of elements of \(G\) and induces the restriction map on cohomology groups, and cor is the augmentation map of the group algebra and induces the corestriction map on cohomology groups. By decomposing this exact sequence and a dual exact sequence of \(\Gamma\)- modules into short exact sequences and taking maps and composites of maps from the resulting long exact sequences of cohomology, the authors obtain the following complexes: \[ \begin{multlined} S_ n: H^ n M\oplus (H^ n F)^ 3 \to \bigoplus^ 3_{i=1} H^ n L_ i\to (H^ n F)^ 2\to H^{n+1} F\overset\text{res}\longrightarrow\\ \overset \text{res} \longrightarrow H^{n+1} {M} \bigoplus^…
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 13D25
zbMATH DE Number: 218291
zbMATH Keywords: Galois group
zbMATH Keywords: biquadratic extension
zbMATH Keywords: exact sequence
zbMATH Keywords: homology groups
zbMATH Keywords: complexes
zbMATH Keywords: Galois cohomology
author: Alexander Merkurjev
author: Jean-Pierre Tignol
reviewed by: H. F. Kreimer
MaRDI profile type: Publication
OpenAlex ID: W2005134354
description: scientific article; zbMATH DE number 218291