On the cohomology of biquadratic extensions
type: Item
instance of: scholarly article
title: On the cohomology of biquadratic extensions
zbMATH Open document ID: 0798.12007
DOI: 10.1007/BF02564476
published in: Commentarii Mathematici Helvetici
publication date: 1994-10-31T00:00:00Z
full work available at URL: https://eudml.org/doc/140328
review text: Let \(F\) be a field of characteristic \(\neq 2\) and \(E/F\) a biquadratic Galois extension. Relationships between the Galois cohomology (with coefficients \(\mathbb{Z}/ 2\mathbb{Z}\)) of \(F\), \(E\) and the three intermediate quadratic extensions have been studied by various authors. In particular, Merkurjev and Tignol have introduced two series of seven-term complexes \(S_ n\) and \(S^ n\) \((n\geq 0)\) involving these cohomology groups, but too technical to be written down here, and have proved their exactness in some cases, e.g. for \(n\leq 1\). The main result of this paper is that \(S_ n\) and \(S^ n\) are exact sequences in the following cases: (1) \(n=2\). (2) The Milnor conjecture holds for \(F\) and all its finite extensions in degree \(\leq n\), and a) \(F= F^ 2+ F^ 2\) or b) \(E\) is Pythagorean. Recall that \(K\) is Pythagorean if the set of squares of \(K\) is closed under addition, and that the Milnor conjecture in degree \(n\) for \(K\) predicts that the natural homomorphism \(K_ n^ M(K)/2\to H^ n(K, \mathbb{Z}/2)\) is an isomorphism.
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 590900
zbMATH Keywords: biquadratic Galois extension
zbMATH Keywords: Galois cohomology
zbMATH Keywords: seven-term complexes
zbMATH Keywords: exact sequences
zbMATH Keywords: Milnor conjecture
zbMATH Keywords: Pythagorean
author: Bruno Kahn
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 590900