Springer's theorem for tame quadratic forms over Henselian fields
type: Item
instance of: scholarly article
title: Springer's theorem for tame quadratic forms over Henselian fields
zbMATH Open document ID: 1270.11039
published in: Mathematische Zeitschrift
publication date: 2011-10-25T00:00:00Z
full work available at URL: https://arxiv.org/abs/1002.1153
review text: A famous theorem by T.~A.~Springer states that the Witt group of quadratic forms over a complete discretely valued field of residue characteristic not \(2\) is isomorphic to a direct sum of two copies of the Witt group of the residue field. The case of residue characteristic \(2\) poses many subtle problems. The purpose of the present paper is to obtain a very general and characteristic free version of Springer's theorem for a certain subgroup of the Witt group which the authors call the Witt group of tame quadratic forms over a Henselian field \(F\) with valuation \(v\), denoted by \(I_{qt}(F)\). More precisely, let \(I_q(F)\) be the Witt group of even-dimensional nondegenerate quadratic forms over \(F\). Then \(I_{qt}(F)\) is the subgroup consisting of those classes of forms that become hyperbolic over a tame extension of \(F\), where a finite degree field extension \(E\) of the Henselian field \(F\) is called tame if it is defectless, with ramification degree not divisible by the characteristic of the residue field \(\overline{F}\) and its residue field \(\overline{E}\) is a separable extension of \(\overline{F}\). The paper's main theorem states that there is an isomorphism \(\partial: I_{qt}(F)\to \bigoplus_{\Gamma/2\Gamma}I_q(\overline{F})\) where \(\Gamma\) denotes the value group of \(v\), and where \(\partial\) is defined by residue forms for each coset of \(\Gamma\) modulo \(2\Gamma\) that depend on a choice of uniformizing parameter for the respective coset. This result in the case \(\text{char}(\overline{F})=2\) was essentially known and due to \textit{U.-P. Tietze} [Arch. Math. 25, 144--150 (1974; Zbl 0285.15018)], but his formulation and methods were quite different as he worked with the subgroup \(U(F)\) of \(I_q(F)\) generated by forms \(ax^2+bxy+cy^2\) with \(v(a)=v(b)=v(c)=1\) and used generators and relations for the Witt group. The authors show in the present paper that in fact \(U(F)=I_{qt}(F)\), thus recovering Tietze's result when \(\mathrm{ch…
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 5963445
zbMATH Keywords: quadratic form
zbMATH Keywords: Witt group
zbMATH Keywords: Henselian valuation
zbMATH Keywords: tame extension
author: Mohamed Abdou Elomary
author: Jean-Pierre Tignol
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
OpenAlex ID: W1987613571
arXiv ID: 1002.1153
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DOI: 10.1007/S00209-010-0729-Y
description: scientific article