Witt rings under odd degree extensions
type: Item
instance of: scholarly article
title: Witt rings under odd degree extensions
zbMATH Open document ID: 0790.11034
author: Robert W. Fitzgerald
published in: Pacific Journal of Mathematics
publication date: 1994-06-26T00:00:00Z
review text: Let \(K\) be a separable odd degree extension of the field \(F\). Then the Witt ring \(WK\) can be viewed as a \(WF\)-module and, on the other hand, the kernel \(\ker s_ *\) of the Scharlau transfer \(s_ *: WK\to WF\) is also a \(WF\)-module. The main theme of the paper is to decide whether or not these are projective or free \(WF\)-modules. Since \(WF\) need not be Noetherian and \(\ker s_ *\) need not be finitely generated, a generalized theory of attached primes is used to answer the main question. It turns out that \(WK\) is not a projective \(WF\)-module if some ordering on \(F\) extends uniquely to \(K\). An example is produced to show that \(WK\) may be a free \(WF\)-module if \(F\) is real and no ordering extends uniquely. When \(F\) is nonreal and nonrigid, then \(WK\) is not a projective \(WF\)-module provided \(K/F\) is Galois and \(WK\) is Noetherian.
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 509099
zbMATH Keywords: Witt ring
zbMATH Keywords: Scharlau transfer
zbMATH Keywords: projective \(WF\)-module
reviewed by: Kazimierz Szymiczek
MaRDI profile type: Publication
full work available at URL: pjm.1993.158.121
OpenAlex ID: W2028814368
DOI: 10.2140/PJM.1993.158.121
description: scientific article; zbMATH DE number 509099