On quadratic forms whose total signature is zero mod \(2^n\). Solution to a problem of M. Marshall
type: Item
instance of: scholarly article
title: On quadratic forms whose total signature is zero mod \(2^n\). Solution to a problem of M. Marshall
zbMATH Open document ID: 0908.11022
published in: Inventiones Mathematicae
publication date: 1998-11-29T00:00:00Z
review text: The main aim of the paper is to prove that if \(F\) is a formally real pythagorean field and the signature sgn\(_P(\phi)\) of the quadratic form \(\phi\) over \(F\) is zero modulo \(2^n\) for every order \(P\) of \(F\), then \(\phi\in I^n(F)\), where \(I(F)\) is the fundamental ideal of the Witt ring \(W(F)\). It solves the problem posed by \textit{M. Marshall} [|Proc. conf. on quadratic forms, Queen's Pap. Pure Appl. Math. 46, 569-579 (1977; Zbl 0385.12008)]. The paper is divided into two parts. In the first part the authors reformulate the original problem into the so-called Weak Marshall Conjecture, which is given an affirmative answer in the second part. The main result depends on Voevodsky's recent result on Milnor homomorphisms. In the first part the authors work in the framework of so called special groups and related Boolean-theoretic techniques. The theory of special groups is introduced in their forthcoming paper and resembles a modified version of Marshall's abstract theory of quadratic forms. For the convenience of the reader, they recall the basic notions and facts of this theory.
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 1199914
zbMATH Keywords: formally real fields
zbMATH Keywords: pythagorean fields
zbMATH Keywords: Witt ring
zbMATH Keywords: special groups
zbMATH Keywords: Boolean homomorphisms
zbMATH Keywords: Boolean hull
zbMATH Keywords: inductive graded ring of exponent two
author: Max A. Dickmann
author: Francisco Miraglia
reviewed by: Andrzej Sładek
MaRDI profile type: Publication
full work available at URL: s002220050246
OpenAlex ID: W376655269
DOI: 10.1007/S002220050246
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description: scientific article; zbMATH DE number 1199914