The root number class and Chinburg's second invariant
type: Item
instance of: scholarly article
title: The root number class and Chinburg's second invariant
zbMATH Open document ID: 0769.11041
DOI: 10.1016/0021-8693(92)90152-C
author: Seyong Kim
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: Let \(N/K\) be a finite normal extension of number fields, \(G=\text{Gal}(N/K)\), \(D=\mathbb{Z}[G]\). The ring of integers \({\mathcal O}_ N\) defines a class \(\Omega_ a(N/K)=({\mathcal O}_ N)-[K: \mathbb{Q}](D)\) in the class-group \(C\ell(D)\) (= the finite torsion subgroup of \(K_ 0(D))\). In Invent. Math. 63, 41-79 (1981; Zbl 0469.12003), \textit{M. J. Taylor} proved Fröhlich's conjecture, that \(\Omega_ a(N/K)\) is equal to the root number class \(W_{N/K}\) defined by means of the Artin root numbers of the irreducible symplectic representations of \(G\). Moreover, in Ann. Math., II. Ser. 121, 351-376 (1985; Zbl 0567.12010), \textit{T. Chinburg} defined Galois invariants \(\Omega(N/K,i)\), \(i=1,2,3\) in \(C\ell(D)\) and proved that \(\Omega(N/K,2)=\Omega_ a(N/K)\) for tamely ramified \(N/K\). He also conjectured that \(\Omega(N/K,2)=W_{N/K}\) for all \(N/K\). In the present paper, the author proves Chinburg's conjecture in the case where \(K=\mathbb{Q}\), \(G\simeq\mathbb{H}_ 8\) and 2 is not totally ramified in \(N\).
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 130472
zbMATH Keywords: ring of integers
zbMATH Keywords: class-group
zbMATH Keywords: root number class
zbMATH Keywords: Galois invariants
zbMATH Keywords: Chinburg's conjecture
MaRDI profile type: Publication
cites work: Q5521595
cites work: Q3240881
cites work: The analytic theory of multiplicative Galois structure
cites work: Exact sequences and Galois module structure
cites work: Q3966247
cites work: Artin Root Numbers, Conductors, and Representations for Generalized Quaternion Groups
cites work: Artin-root numbers and normal integral bases for quaternion fields
cites work: A generalization of Froehlich's theorem to wildly ramified quaternion extensions of \(\mathbb Q\)
cites work: Q4132638
cites work: Q4132637
cites work: Modules sur l'algèbre du groupe quaternionien
cites work: Q4061092
cites work: Q3856855
cites work: On Fröhlich's conjecture for rings of integers of tame extensions
Recommended article: On Chinburg's root number conjecture
Recommended article: Q3419512
Recommended article: On Root Invariants of Periodic Classes in Ext A (Z/2,Z/2)
Recommended article: Q4855625
Recommended article: Q3829632
Recommended article: Root numbers of elliptic curves in residue characteristic 2
Recommended article: Q4400701
Recommended article: An analogue of Artin's primitive root conjecture
Recommended article: The second Chinburg conjecture for quaternion fields
Recommended article: Cyclotomic Galois module structure and the second Chinburg invariant
description: scientific article; zbMATH DE number 130472