Witt groups of algebraic integers
type: Item
instance of: scholarly article
title: Witt groups of algebraic integers
zbMATH Open document ID: 0651.10014
DOI: 10.1016/0022-314X(88)90001-7
author: Parvati Shastri
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: Let \(F\) be an algebraic number field and let \(D\) be its ring of integers. There is an injective group homomorphism \(WD\to WF\), where \(WD\) and \(WF\) are Witt rings of \(D\) and \(F\), respectively. The author is interested in determining the structure of the additive group \(WD\) and in deciding when \(WD\to WF\) splits. He determines the structure of the abelian group \(WD\) in terms of arithmetic invariants of the field \(F\). He also has some necessary and sufficient conditions for the splitting problem. For instance, if \(F\) is totally imaginary, then the condition is that every element of \(WD\) 2-divisible in \(WF\) be 2-divisible in \(WD\). For real quadratic number fields the condition is given in terms of prime factors of the discriminant, and for imaginary quadratic fields it is shown that \(WD\to WF\) always splits. \{In case of quadratic number fields there is an overlap with two papers written at the University of Bielefeld by \textit{W. Colberg} [Wittringe von Dedekindringen, Diplomarbeit, 1975] and \textit{R. Münstermann} [Der Wittring des Ringes der ganzen Zahlen eines quadratischen Zahlkörpers, Staatsexamensarbeit, 1983]. Colberg determines the additive structure of \(WD\) for any quadratic number field \(F\), and Münstermann, in addition to determining the generators of direct summands in a decomposition of \(WD\) for a quadratic field \(F\), shows how to multiply the generators thus describing completely the ring structure of \(WD\).\}
reviewed by: Kazimierz Szymiczek
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 11E12
zbMATH DE Number: 4061323
zbMATH Keywords: algebraic number field
zbMATH Keywords: Witt rings
zbMATH Keywords: splitting
zbMATH Keywords: totally imaginary
zbMATH Keywords: real quadratic number fields
zbMATH Keywords: imaginary quadratic fields
MaRDI profile type: Publication
cites work: Q5521595
cites work: Q5532660
cites work: Q5577214
cites work: Q3856844
cites work: Q5829819
cites work: Q5675788
cites work: Q4043127
cites work: Q4402415
cites work: Q5607309
cites work: Q3708857
full work available at URL: 0022-314x(88)90001-7
OpenAlex ID: W2024361895
description: scientific article; zbMATH DE number 4061323