Composition algebras over a ring of fractions
type: Item
instance of: scholarly article
title: Composition algebras over a ring of fractions
zbMATH Open document ID: 0887.17003
author: Susanne Pumplün
published in: Journal of Algebra
publication date: 1998-05-27T00:00:00Z
review text: Let \(k\) be a field of characteristic distinct from 2, and let \(R\) be the ring of fractions defined by \[ R=\left\{{g(t) \over f(t)^j} \in k(t) \mid j\geq 0,\;g(t)\in k[t] \text{ of }\deg \leq 2j \right\} \simeq k[x_0,x_1]_{(x^2_0 -ax_1^2)} \] with \(f(t)=t^2- a\in k[t]\) and \(a\in k^\times\). The author considers composition algebras without zero divisors over the ring \(R\), lists and partly classifies them. She gives also some examples for special base fields \(k\). Her main results are: The Picard group of a nonsplit torus defined over \(k\) contains at most two elements, one of them the torus itself considered as an \(R\)-module. Every nonsplit quaternion algebra over \(R\) not defined over \(k\) can be realized as a nonclassical Cayley-Dickson doubling of a torus defined over \(k\) or, equivalently, as a classical Cayley-Dickson doubling of a torus not defined over \(k\). Octonion algebras not defined over \(k\) can either be realized as a nonclassical Cayley-Dickson doubling of some quaternion algebra, or, equivalently, as a classical Cayley-Dickson doubling of a quaternion algebra not defined over \(k\).
reviewed by: Huberta Lausch
Mathematics Subject Classification ID: 17A75
zbMATH DE Number: 1014020
zbMATH Keywords: ring of fractions
zbMATH Keywords: octonion algebras
zbMATH Keywords: composition algebras without zero divisors
zbMATH Keywords: Picard group
zbMATH Keywords: nonsplit quaternion algebra
zbMATH Keywords: Cayley-Dickson doubling
MaRDI profile type: Publication
full work available at URL: jabr.1996.6809
OpenAlex ID: W1976995858
cites work: Quadratic forms permitting composition
cites work: Q3994498
cites work: Q3264270
cites work: Q4143433
cites work: Composition algebras and their automorphisms
cites work: Q5670744
cites work: Q3997794
cites work: Q4314102
cites work: Q3791303
cites work: Serre's conjecture
cites work: Q5675788
cites work: Nonassociative algebras with scalar involution
cites work: Q3871006
cites work: Composition Algebras Over Algebraic Curves of Genus Zero
cites work: Composition algebras over a field with a discrete valuation
cites work: Composition algebras over rings of genus zero
cites work: Q3708857
DOI: 10.1006/JABR.1996.6809
description: scientific article; zbMATH DE number 1014020