Rings of integers and trace forms for tame extensions of odd degree
type: Item
instance of: scholarly article
title: Rings of integers and trace forms for tame extensions of odd degree
zbMATH Open document ID: 0723.11055
DOI: 10.1007/BF01159963
author: Martin J. Taylor
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174094
review text: Set \(\Gamma =Gal(N/K)\) where N/K is a tame Galois extension of number fields of odd degree. Let R denote the ring of integers of N and \(t_ K: N\times N\to K\Gamma\) be the Hermitian form associated with the trace map for N/K. The main goal of this article is to determine the image, under restriction, of the class of the Hermitian pair \((R,t_ K)\) in the Grothendieck group of locally free \(Z\Gamma\)-modules. A characterization is given in terms of generalized Swan modules which generalize the Lagrangian bases that were determined by \textit{P. E. Conner} and \textit{R. Perlis} [A survey of trace forms of algebraic number fields (1984; Zbl 0551.10017)]. The Artin root numbers play a crucial role in the proof of the main theorem. Also, the proof gives rise to the question of a possible converse result: namely, does the class determine the square of the root numbers of all characters of \(\Gamma\) ?
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R54
Mathematics Subject Classification ID: 11E39
zbMATH DE Number: 4191815
zbMATH Keywords: tame Galois extension
zbMATH Keywords: ring of integers
zbMATH Keywords: Hermitian form
zbMATH Keywords: trace map
zbMATH Keywords: generalized Swan modules
zbMATH Keywords: Artin root numbers
reviewed by: Charles J. Parry
MaRDI profile type: Publication
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OpenAlex ID: W2027247347
description: scientific article; zbMATH DE number 4191815