Local and global fundamental classes for multiquadratic extensions
type: Item
instance of: scholarly article
title: Local and global fundamental classes for multiquadratic extensions
zbMATH Open document ID: 1283.11156
author: Paul Buckingham
published in: Journal of Number Theory
publication date: 2013-01-10T00:00:00Z
review text: Fundamental classes play a dominant role in algebraic number theory and particularly in class field theory. Given an extension \(L/K\) with Galois group \(G\), the fundamental class is given in the local case by an element of H\(^2(G,L^\times)\) and in the global case by an element of H\(^2(G,C_L)\), with \(C_L\) the idele class group of \(L\). In both cases the fundamental class is realised by a certain 2-extension, in other words, a 4-term exact sequence whose two middle terms are \(G\)-cohomologically trivial (c.t.). These classes and sequences are hard to determine explicitly. The local case with cyclic Galois group \(G\) is fairly transparent, since then the H\(^2\) term can be replaced by a H\(^0\) term (\(G\)-fixed elements modulo norms), which is way easier to handle. The paper under review successfully deals with a case which is simple in some way but very far from being cyclic: \(L/K\) is a compositum of \(r\) quadratic extensions. The starting point of this very nice paper is a description of the relevant 4-term sequence in the local case, due to Serre. One has two short exact sequences, both with c.t. middle term \((\hat K_{ur}\otimes L)^\times\); the left hand term of the former is \(L^\times\) and the right hand term of the latter is \(\mathbb Z\). Moreover, the right hand term \(V\) of the former exact sequence agrees with the left hand term of the latter, so that one may concatenate them, to form a 2-extension (four-term exact sequence) leading from \(L^\times\) to \(\mathbb Z\), the two modules in the middle being identical and c.t. (There is a harmless typo in the paper: it is not \(V\) which is c.t. but the module \((\hat K_{ur}\otimes L)^\times\), as is clearly indicated by the context.) This 2-extension is reworked somewhat before being used in the determination of the fundamental class. We do not go into detail and only give the final outcome in the simpler of two cases (\(L/K\) is totally ramified): Let \(G\) be 2-elementary of order \(2^r\) w…
reviewed by: Cornelius Greither
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11S31
zbMATH DE Number: 6124186
zbMATH Keywords: Fundamental classes
zbMATH Keywords: reciprocity map
zbMATH Keywords: class field theory
zbMATH Keywords: multiquadratic extension
MaRDI profile type: Publication
full work available at URL: j.jnt.2012.08.014
OpenAlex ID: W2063215568
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DOI: 10.1016/J.JNT.2012.08.014
description: scientific article; zbMATH DE number 6124186