Galois cohomology and the trace form
type: Item
instance of: scholarly article
title: Galois cohomology and the trace form
zbMATH Open document ID: 0804.12003
published in: Jahresbericht der Deutschen Mathematiker-Vereinigung (DMV)
publication date: 1995-01-23T00:00:00Z
review text: This is a very readable survey on recent progress concerning trace forms and Galois cohomology. Let \(K\) be a field of characteristic \(\neq 2\), let \(f\in K[X]\) be a separable polynomial and let \(L= K[X]/ (f)\). The trace form \(q_ L: L\to K\), defined by \(q_ L(x):= \text{Tr}_{L/K} (x^ 2)\), is a nondegenerate quadratic form over the \(K\)-vector space \(L\). The author first reviews results from early history of trace forms, cohomological invariants as quadratic form to recent results of Serre (unpublished) on cohomological invariants of an étale algebra, as well as applications of the vanishing formula for the Stiefel-Whitney classes, on solvability of certain Galois embedding problems. Results concerning which quadratic forms are trace forms are also considered. The last part is devoted to trace forms of Galois extensions, the aim being to describe the isomorphism class of \(q_ L\) as \(G\)-form, if \(L/K\) is a Galois extension with Galois group \(G\), in terms of cohomological invariants.
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 12-02
zbMATH DE Number: 566045
zbMATH Keywords: trace form
zbMATH Keywords: Galois cohomology
zbMATH Keywords: cohomological invariants
zbMATH Keywords: Stiefel-Whitney classes
zbMATH Keywords: Galois embedding problems
zbMATH Keywords: quadratic forms
zbMATH Keywords: Galois extensions
author: Eva Bayer-Fluckiger
reviewed by: Nuria Vila
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 566045