Galois algebras, Hasse principle and induction-restriction methods
type: Item
instance of: scholarly article
title: Galois algebras, Hasse principle and induction-restriction methods
zbMATH Open document ID: 1247.11046
published in: Documenta Mathematica
publication date: 2011-12-09T00:00:00Z
full work available at URL: 24.html
review text: Let \(L/k\) be a Galois field extension with group \(G\). It is well known that \(L/k\) has a \textit{normal basis}, i.e., a \(k\)-vector space basis which consists of a single \(G\)-orbit in \(L\). Assume \(\mathrm{char}(k) \neq 2\). To \(L/k\) we can associate its trace form \(q_L: L\times L\to k\), defined by \(q_L(x,y)=\mathrm{Tr}_{L/k}(xy)\). It is a \(G\)-quadratic form on \(L\), i.e., satisfies \(q_L(gx,gy)=q_L(x,y)\) for all \(g\in G\) and \(x,y\in L\). The question arises whether \(L/k\) has a normal basis which satisfies \(q_L(gx,hx)=\delta_{g,h}\) for all \(g,h\in G\) and \(x\in L\). Such a normal basis is called \textit{self-dual}. This problem has been considered in [\textit{E. Bayer-Fluckiger} and \textit{H.W. Lenstra, jun.}, Am. J. Math. 112, No. 3, 359--373 (1990; Zbl 0729.12006)], where a positive answer is given in case \([L:k]\) is odd, and in [\textit{E. Bayer-Fluckiger} and \textit{J.-P. Serre}, Am. J. Math. 116, No. 1, 1--64 (1994; Zbl 0804.12004)], which contains partial results in case \([L:k]\) is even. The focus of the paper under review is on the following Hasse principle type of question for global fields \(k\): Suppose that \(L\) and \(L'\) are \(G\)-Galois algebras (e.g. \(G\)-Galois field extensions) over \(k\). Assume that for all places \(\nu\) of \(k\) the \(G\)-forms \(q_{L_\nu}\) and \(q_{L'_\nu}\) are isomorphic over the completion \(k_\nu\). Are the \(G\)-forms \(q_L\) and \(q_{L'}\) isomorphic over \(k\)? Note that \(L/k\) has a self-dual normal basis if and only if \(q_L\) is isomorphic as a \(G\)-quadratic form to the unit form \(u=q_{k[G]}\) on the group algebra \(k[G]\), determined by \(u(g,h)=\delta_{g,h}\). A key ingredient for analyzing this question is the \textit{odd determinant property} of a finite group \(G\) introduced in this paper. Choose a \(2\)-Sylow subgroup \(S\) of \(G\) and let \(X\) be the free abelian group of \(\mathbb{Z}\)-valued functions on \(S\) invariant under conjugation by the normalizer \(N=N_{G}…
reviewed by: Roland Lötscher
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 5987891
zbMATH Keywords: Galois algebra
zbMATH Keywords: Hasse principle
zbMATH Keywords: induction
zbMATH Keywords: restriction
zbMATH Keywords: \(G\)-quadratic space
zbMATH Keywords: trace form
zbMATH Keywords: normal basis
author: Eva Bayer-Fluckiger
author: R. Parimala
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5987891